a quasi monte carlo method for the boltzmann equation a new quasi monte carlo method for solving the boltzmann equation in a simplified case is described the analysis is restricted to a spatially homogeneous and isotropic gas in addition the molecular model only involves isotropic scattering the scheme makes use of particles and combines an euler scheme with numerical integrations the sequence which is used for the quadratures must possess some symmetry properties which prescribe energy conservation for colliding particles the error of the method is estimated by means of the discrepancy of the sequence which performs the quadratures an algorithm for generating convenient sequences is proposed in an example where an exact solution is known the computation of effective errors is included introduction rarefied gas flows are usually simulated by monte carlo techniques besides the successful direct simulation monte carlo dsmc method of bird 2 another scheme was derived by nanbu 14 from the boltzmann equation itself a drawback of both schemes are numerical fluctuations caused by the use of pseudorandom numbers an improved monte carlo scheme which reduces fluctuations has recently been developed at the university of kaiserslautern 1 it will be referred to as the kmc scheme a fully deterministic method for solving the boltzmann equation is proposed here the constraints on the analysis are the following we consider an infinite spatially homogeneous and isotropic gas the velocity distribution is radially symmetric the molecular model is characterized by isotropic scattering the differential cross section a depends only on the relative speed g and not on the deflection angle in addition we assume that ga g is some nonnegative nondecreasing and bounded function the hypothesis on the cross section allows physically relevant models such as the vhs model of bird 3 when a cutoff is used in earlier communications io i i the simplest choice ga g equals a constant was considered a deterministic version of the scheme of nanbu that we called the low discrepancy ld method was described an error analysis was proposed and it was shown that the ld method outperforms the original scheme of nanbu in a test case where an exact solution is known received march 1 1990 1980 mathematics subject classification 1985 revision primary 65m15 secondary 65c05 llk38 1991 american mathematical society 0025 5718 91 l 00 25 per page

stable particle like solutions to the nonlinear wave equations of three dimensional excitable media so called excitable media support nonlinear waves that propagate as pulses e g the action potential in nerve fibers and heart muscle and the oxidation pulse in certain chemical reactions in two dimensional media the right stimulus will also create rotating vortices which radiate such pulses periodically in every direction their investigation in diverse excitable media has dramatically expanded in the usa and ussr since their discovery in both countries in 1970 in three dimensional media the pivot of the rotating structure is not a point but a curved line generically a closed ring rings linked together and knotted constitute topologically distinctive organizing centers for periodic activity throughout the medium it is of interest to understand how stimuli create organizing centers and how organizing centers persist or decay the numerically discovered stable organizing centers exhibit enough symmetries to encourage mathematical attention this paper illustrates organizing centers by computer g

a quasi monte carlo approach to particle simulation of the heat equation the convergence of the monte carlo method for numerical integration can often be improved by replacing random numbers with more uniformly distributed numbers known as quasi random in this paper the convergence of monte carlo particle simulation is studied when these quasi random sequences are used for the one dimensional heat equation discretized in both space and time convergence is proved for a quasi random simulation using reordering of the particles according to their position experimental results are presented for the spatially continuous heat equation in one and two dimensions the results indicate that a significant improvement in both magnitude of error and convergence rate can be achieved over standard monte carlo simulations for certain low dimensional problems

path integral calculation of the writhe for circular semiflexible polymers earlier obtained numerical and analytical non path integral results for the writhe of a circular polymer are reproduced exactly employing the available path integral models for the semiflexible chains it is confirmed analytically that the average writhe scales with the root of the chain length we also briefly discuss related topological problems e g knotting of circular polymers which could be formulated and solved with the use of path integrals

the direct link model for polymer rings using topological field theory and the second topological moment in dense systems polymer rings in solution are either permanently entangled or are not permanent topological restrictions give rise to additional entropic interactions apart from the ones arising due to mere chain flexibility or excluded volume conversely entangled polymer rings systems may be formed by closing randomly entangled flexible linear chains the dependence of linking numbers between randomly entangled rings on the chain length more specifically the second topological moment n2 i e the average squared linking number may be determined in this paper an approach recently discussed in mathematical physics and called abelian bf theory is presented which allows one to express the linking constraint in its simplest form the gauss integral in terms of two gauge fields the model of brereton and shah for a single ring entangled with many other surrounding rings is rederived the latter model is finally used to calculate the second topological moment in agreement with a recent result by ferrari kleinert and lazzizzera obtained using n component 4 theory for the limit n 0

topology bounds energy of knots and links in this paper we determine two quantities of geometric and topological character that were left undetermined in two previous results obtained by arnold arnold 1974 in proc summer school in dif

fast generalized minimum distance decoding of algebraic geometry and reed solomon codes generalized minimum distance gmd decoding is a standard soft decoding method for block codes we derive an efficient general gmd decoding scheme for linear block codes in the framework of error correcting pairs special attention is paid to reed solomon rs codes and one point algebraic geometry ag codes for rs codes of length n and minimum hamming distance d the gmd decoding complexity turns out to be in the order o nd where the complexity is counted as the number of multiplications in the field of concern for ag codes the gmd decoding complexity is highly dependent on the curve in consideration it is shown that we can find all relevant error erasure locating functions with complexity o o sub 1 nd where o sub 1 is the size of the first nongap in the function space associated with the code a full gmd decoding procedure for a one point ag code can be performed with complexity o dn sup 2

static and dynamic stability results for a class of three dimensional configurations of kirchhoff elastic rods we analyze the dynamical stability of a naturally straight inextensible and unshearable elastic rod under tension and controlled end rotation within the kirchhoff model in three dimensions the cases of clamped boundary conditions and isoperimetric constraints are treated separately we obtain explicit criteria for the static stability of arbitrary extrema of a general quadratic strain energy we exploit the equivalence between the total energy and a suitably defined norm to prove that local minimizers of the strain energy under explicit hypotheses are stable in the dynamic sense due to liapounov we also extend our analysis to damped systems to show that static equilibria are dynamically stable in the liapounov sense in the presence of a suitably defined local drag force

hamiltonians and variational principles for alfven simple waves the evolution equations for the magnetic field induction b with the wave phase for alfven simple waves are expressed as variational principles and in the hamiltonian form the evolution of b with the phase which is a function of the space and time variables depends on the generalized frenet serret equations in which the wave normal n which is a function of the phase is taken to be tangent to a curve x in a 3d cartesian geometry vector space the physical variables the gas density fluid velocity gas pressure and magnetic field induction in the wave depend only on the phase three approaches are developed one approach exploits the fact that the frenet equations may be written as a 3d hamiltonian system which can be described using the nambu bracket it is shown that b as a function of the phase satisfies a modified version of the frenet equations and hence the magnetic field evolution equations can be expressed in the hamiltonian form a second approach develops an euler poincare variational formulation a third approach uses the frenet frame formulation in which the hodograph of b moves on a sphere of constant radius and uses a stereographic projection transformation due to darboux the equations for the projected field components reduce to a complex riccati equation by using a cole hopf transformation the riccati equation reduces to a linear second order differential equation for the new variable a hamiltonian formulation of the second order differential equation then allows the system to be written in the hamiltonian form alignment dynamics equations for alfven simple waves give rise to a complex riccati equation or equivalently to a quaternionic riccati equation which can be mapped onto the riccati equation obtained by stereographic projection

towards a classification of euler kirchhoff filaments euler kirchhoff filaments are solutions of the static kirchhoff equations for elastic rods with circular cross sections these equations are known to be formally equivalent to the euler equations for spinning tops this equivalence is used to provide a classification of the different shapes a filament can assume explicit formulas for the different possible configurations and specific results for interesting particular cases are given in particular conditions for which the filament has points of self intersection self tangency vanishing curvature or when it is closed or localized in space are provided the average properties of generic filaments are also studied they are shown to be equivalent to helical filaments on long length scales

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

balancing torques in membrane mediated interactions exact results and numerical illustrations torques on interfaces can be described by a divergence free tensor which is fully encoded in the geometry this tensor consists of two terms one originating in the couple of the stress the other capturing an intrinsic contribution due to curvature in analogy to the description of forces in terms of a stress tensor the torque on a particle can be expressed as a line integral along a contour surrounding the particle interactions between particles mediated by a fluid membrane are studied within this framework in particular torque balance places a strong constraint on the shape of the membrane symmetric two particle configurations admit simple analytical expressions which are valid in the fully nonlinear regime in particular the problem may be solved exactly in the case of two membrane bound parallel cylinders this apparently simple system provides some flavor of the remarkably subtle nonlinear behavior associated with membrane mediated interactions

cyclic rotations contractibility and gauss bonnet let a rigid object or frame of reference have identical initial and final orientations but be rotated in any way in between with an angular velocity any unit vector carried with the frame passes through a cycle of directions enclosing a solid angle the full relation between these three quantities is shown to be where the turn number n is zero if the sequence of orientations of the frame is contractible and unity if it is non contractible the main derivation uses the calugareanu relation lk w r tw between linking number writhe and twist of a ribbon loop an outline alternative derivation uses the berry phase of a quantum spin finally the result is applied to the standard parallel transport holonomy expressed in the gauss bonnet theorem it is refined to be correct rather than merely

the fourier transform of tubular densities we consider the fourier transform of tubular volume densities with arbitrary axial geometry and possibly twisted internal structure this density can be used to represent among others magnetic flux or the electron density of biopolymer molecules we consider tubes of both finite radii and unrestricted radius when there is overlap of the tube structure the net density is calculated using the super position principle the fourier transform of this density is composed of two expressions one for which the radius of the tube is less than the curvature of the axis and one for which the radius is greater which must have density overlap this expression can accommodate an asymmetric density distribution and a tube structure which has non uniform twisting in addition we give several simpler expressions for isotropic densities densities of finite radius densities which decay at a rate sufficient to minimise local overlap and finally individual surfaces of the tube manifold these simplified cases can often be expressed as arclength integrals and can be evaluated using a system of first order ode s

on the average complexity of reed solomon list decoders the number of monomials required to interpolate a received word in an algebraic list decoder for reed solomon codes depends on the instantaneous channel error and not only on the decoder design parameters the implications of this fact are that the decoder should be able to exhibit lower decoding complexity for low weight errors and consequently enjoy a better average case decoding complexity and a higher decoding throughput on the analytical side this paper studies the dependence of interpolation costs on instantaneous errors in both hard and soft decision decoders on the algorithmic side it provides an efficient interpolation algorithm based on the state of the art interpolation algorithm that enjoys reduced running times for reduced interpolation costs

isoperimetric problems for the helicity of vector fields and the biot savart and curl operators the helicity of a smooth vector field defined on a domain in three space is the standard measure of the extent to which the field lines wrap and coil around one another it plays important roles in fluid mechanics magnetohydrodynamics and plasma physics the isoperimetric problem in this setting is to maximize helicity among all divergence free vector fields of given energy defined on and tangent to the boundary of all domains of given volume in three space the biot savart operator starts with a divergence free vector field defined on and tangent to the boundary of a domain in three space regards it as a distribution of electric current and computes its magnetic field restricting the magnetic field to the given domain we modify it by subtracting a gradient vector field so as to keep it divergence free while making it tangent to the boundary of the domain the resulting operator when extended to the l2 completion of this family of vector fields is compact and self adjoint and thus has a largest

analytical expression of elastic rods at equilibrium under 3d strong anchoring boundary conditions abstract a general purpose method is presented and implemented to express analytically one stationary configuration of an ideal 3d elastic rod when the end to end relative position and orientation are imposed the mechanical equilibrium of such a rod is described by ordinary differential equations and parametrized by six scalar quantities when one end of the rod is anchored the analytical integration of these equations lead to one unique solution for given values of these six parameters when the second end is also anchored six additional nonlinear equations must be resolved to obtain parameter values that fit the targeted boundary conditions we find one solution of these equations with a zero finding algorithm by taking initial guesses from a grid of potential candidates we exhibit the symmetries of the problem which reduces drastically the size of this grid and shortens the time of selection of an initial guess the six variables used in the search algorithm forces and moments at one end of the rod are particularly adapted due to their unbounded definition domain more than 850 000 tests are performed in a large region of configurational space and in 99 9 of cases the targeted boundary conditions are reached with short computation time and a precision better than 10 5 we propose extensions of the method to obtain many solutions instead of only one using numerical continuation or starting from different initial guesses

on the groundstate energy of tight knots new results on the groundstate energy of tight magnetic knots are presented magnetic knots are defined as tubular embeddings of the magnetic field in an ideal perfectly conducting incompressible fluid an orthogonal curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the magnetic field standard minimization of the magnetic energy is carried out under the usual assumptions of volume and flux preserving flow with the additional constraints that the tube cross section remains circular and that the knot length ropelength is independent from internal field twist framing under these constraints the minimum energy is determined analytically by a new exact expression function of ropelength and framing groundstate energy levels of tight knots are determined from ropelength data obtained by the sono tightening algorithm results for torus knots are compared with previous work and the groundstate energy spectrum of the first prime knots up to 10 crossings is presented and analysed in detail these results demonstrate that ropelength and framing determine the spectrum of magnetic knots in tight configuration

a generalization of the berlekamp massey algorithm for multisequence shift register synthesis with applications to decoding cyclic codes a generalization of the berlekamp massey algorithm is presented for synthesizing minimum length linear feedback shift registers for generating prescribed multiple sequences a more general problem is first considered that of finding the smallest initial set of linearly dependent columns in a matrix over an arbitrary field which includes the multisequence problem as a special case a simple iterative algorithm the fundamental iterative algorithm fia is presented for solving this problem the generalized algorithm is then derived through a refinement of the fia application of this generalized algorithm to decoding cyclic codes up to the hartmann tzeng ht bound and roos bound making use of multiple syndrome sequences is considered conditions for guaranteeing that the connection polynomial of the shortest linear feedback shift register obtained by the algorithm will be the error locator polynomial are determined with respect to decoding up to the ht bound and special cases of the roos bound

the jones polynomial for fluid knots from helicity in this paper we prove that under ideal conditions the helicity of fluid knots such as vortex filaments or magnetic flux tubes provides a fundamentally new topological means by which we may associate a topological invariant the jones polynomial that is much stronger than prior interpretations in terms of gauss linking numbers our proof is based on an extension of the kauffman bracket polynomial for unoriented knots explicit calculations of the jones polynomial for the left and right handed trefoil knots and for the whitehead link via the figure of eight knot are presented for illustration this novel approach establishes a topological foundation of classical field theory in general and of mathematical fluid dynamics in particular by opening up new directions of work both in theory and applications

the linking number in systems with periodic boundary conditions periodic boundary conditions pbc are often used for the simulation of complex physical systems using the gauss linking number we define the periodic linking number as a measure of entanglement for two oriented curves in a system employing pbc in the case of closed chains in pbc the periodic linking number is an integer topological invariant that depends on a finite number of components in the periodic system for open chains the periodic linking number is an infinite series that accounts for all the topological interactions in the periodic system in this paper we give a rigorous proof that the periodic linking number is defined for the infinite system i e that it converges for one two and three pbc models it gives a real number that varies continuously with the configuration and gives a global measure of the geometric complexity of the system of chains similarly for a single oriented chain we define the periodic self linking number and prove that it also is defined for open chains in addition we define the cell periodic linking and self linking numbers giving localizations of the periodic linking numbers these can be used to give good estimates of the periodic linking numbers in infinite systems we also define the local periodic linking number associated to chains in the immediate cell neighborhood of a chain in order to study local linking measures in contrast to the global linking measured by the periodic linking numbers finally we study and compare these measures when applied to a pbc model of polyethylene melts

the euler lagrange equation and heat flow for the mobius energy we prove the following results 1 a unique smooth solution exists for a short time for the heat equation associated with the mobius energy of loops in a euclidean space starting with any simple smooth loop 2 a critical loop of the energy is smooth if it has cube integrable curvature combining this with an earlier result of m freedman z wang and the author we show that any local minimizer of the energy must be smooth 3 circles are the only two dimensional critical loops with cube integrable curvature the technique also applies to a family of other knot energies similar problems are open for energies of surfaces or more generally for embedded submanifolds in a fixed riemannian manifold 2000 john wiley sons inc

geometry of călugăreanu s theorem a central result in the space geometry of closed twisted ribbons is clugreanu s theorem also known as white s formula or the clugreanuwhitefuller theorem this enables the integer linking number

a quasi monte carlo scheme using nets for a linear boltzmann equation a quasi monte carlo particle simulation for solving a linear boltzmann equation is constructed and a convergence proof is given the analysis is restricted to the three dimensional equation in the space homogeneous case and the velocity domain is normalized to be i 3 0 1 3 a particle simulation is described it combines a euler scheme in time with quasi monte carlo integration in velocity space the quadratures use 0 m s nets which are sets with a very regular distribution behavior the particles are reordered according to the components of their velocity at each time step a deterministic error bound is obtained finally numerical experiments are presented for some test problems where an exact solution is known the accuracy of the quasi monte carlo scheme is compared with a standard monte carlo algorithm the results show an improvement in both magnitude of error and convergence rate of quasi monte carlo over monte carlo approach

lock on to tape like behaviour in the torsional buckling of anisotropic rods abstract through nonlinear normal form analysis of the equilibrium equations developed in a neighbourhood of the straight rod solution we show the existence of a phase transition from mildly anisotropic to full tape like buckling in non symmetric rods subject to terminal loads provided the material has a sufficiently low poisson s ratio for a solid elliptical rod with a poisson s ratio of 0 2 the critical point is found to occur at a cross sectional aspect ratio of just over 3 for larger aspect ratios the rod locks on to a one twist per wave buckling mode with no internal twist in the mildly anisotropic regime there turn out to be two physically distinct pairs of symmetric so called primary localised buckling modes differing in the phase of the internal twist these four modes homoclinic orbits remain of the full circle of localised buckling modes existing in the symmetric case after breaking of the circular symmetry of the rod s cross section in the strongly anisotropic regime only the energetically favourable pair of flat buckling modes survives

nonlinear euler buckling the buckling of hyperelastic incompressible cylindrical tubes of arbitrary length and thickness under compressive axial load is considered within the framework of nonlinear elasticity analytical and numerical methods for bifurcation are developed using the exact solution of wilkes for the linearized problem within the stroh formalism using these methods the range of validity of the euler buckling formula and its first nonlinear corrections are obtained for third order elasticity the values of the geometric parameters tube thickness and slenderness where a transition between buckling and barrelling is observed are also identified

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

linking twisting writing and helicity on the 2 sphere and in hyperbolic 3 space in the first paper of this series electrodynamics and the gauss linking integral on the 3 sphere and in hyperbolic 3 space we developed a steady state version of classical electrodynamics in these two spaces including explicit formulas for the vector valued green s operator explicit formulas of biot savart type for the magnetic field and a corresponding ampere s law contained in maxwell s equations and then used these to obtain explicit inte gral formulas for the linking number of two disjoint closed curves in this second paper we obtain integral formulas for twisting writhing and helicity and prove that link twist writhe on the 3 sphere and in hyperbolic 3 space we then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces an announcement of these results and a hint of their proofs can be found in the math arxiv math gt 0406276 while an expanded version of the first paper with full proofs can be found at math gt 0510388

entanglement complexity of self avoiding walks self avoiding walks on three dimensional lattices are flexible linear objects which can be self entangled the authors discuss several ways to measure entanglement complexity for n step walks and prove that these complexity measures tend to infinity with n for small n they use monte carlo methods to estimate and compare the n dependence of two of these complexity measures

knotted and linked phase singularities in monochromatic waves exact solutions of the helmholtz equation are constructed possessing wavefront dislocation lines phase singularities in the form of knots or links where the wave function vanishes knotted nothings the construction proceeds by making a nongeneric structure with a strength n dislocation loop threaded by a strength m dislocation line and then perturbing this in the resulting unfolded stable structure the dislocation loop becomes an m n torus knot if m and n are coprime and n linked rings or knots if m and n have a common factor n the loop or rings are threaded by an m stranded helix in our explicit implementation the wave is a superposition of bessel beams accessible to experiment paraxially the construction fails

low discrepancy sequences for solving the boltzmann equation abstract some new quasi monte carlo methods for the solution of the boltzmann equation in a simplified case are described the errors of the methods are estimated by means of the discrepancies of the sequences used for performing the monte carlo quadratures their accuracy is assessed through computation of effective errors in an example where an exact solution is known

divergence free fields energy and asymptotic crossing number in more than one context the equations of classical physics lead to a closed 2 form co on 3 dimensional space which is a conserved quantity in the sense that it pulls back under time evolution in the physics literature co is written as a divergence free vector field w appendix a describes two settings in which a divergence free w is conserved or frozen one is the magnetic field in ideal magneto hydrodynamics and the other is curl velocity in incompressible inviscid fluid flow if ht denotes the transport diffeomorphisms of a moving fluid in which the field is imbedded with ho id wt denotes the field in the fluid at time t and x0 is the space variable at time t 0 then

a particle method for some parabolic equations abstract we present a quasi monte carlo particle simulation of some multidimensional linear parabolic equations with constant coefficients we approximate the elliptic operator in space by a finite difference operator we discretize time into intervals of length t the discrete representation of the solution at time t n n t is a sum of dirac delta measures using the explicit euler scheme the resulting approximation at time t n 1 is recovered by a quasi monte carlo integration we make use of a technique involving renumbering the simulated particles in every time step we state and prove a convergence theorem for the method experimental results are presented for some model problems the results suggest that the quasi monte carlo simulation tends to give more accurate solutions than a monte carlo simulation when the correct renumbering technique is used other choices can result in significant loss of efficiency

supercoiling transitions of closed dna conformations of closed dna molecules are considered within a simple elastic model the interplay of elastic energies of bending and twisting can lead to supercoiling induced by variations of the linking difference these shape transformations are studied in the limit of a large length to thickness ratio of the molecule stationary shapes and energy diagrams are obtained by solving shape equations for closed rings four different families of stationary shapes can be distinguished i planar circles ii nonplanar rings iii self interacting rings and iv interwound configurations they all occur as shapes of minimal energy in the phase diagram of supercoiling the transitions between the different regions in this phase diagram can be either continuous or discontinuous the sequence of shape transitions turns out to be sensitive to the precise values of the elastic parameters the buckling instability of the circle changes within the range of physically accessible values from continuous to discontinuous behavior

extendability of kirchhoff elastic rods in complete riemannian manifolds the kirchhoff elastic rod is a classical mathematical model of equilibrium configurations of thin elastic rods and is defined to be a solution of the euler lagrange equations associated to the energy with the effect of bending and twisting we consider the initial value problem for the euler lagrange equations in a riemannian manifold in a previous paper the author proved the existence and uniqueness of global solutions of the initial value problem in the case where the ambient space is a space form in the present paper we extend this result to the case where the ambient space is a general complete riemannian manifold this implies that an arbitrary kirchhoff elastic rod of finite length in a complete riemannian manifold extends to that of infinite length

kinematic variational principle for motion of vortex rings abstract we show how the ideas of topology and variational principle opened up by euler facilitate the calculation of motion of vortex rings kelvin benjamin s principle as generalised to three dimensions states that a steady distribution of vorticity relative to a moving frame is the state that maximizes the total kinetic energy under the constraint of constant hydrodynamic impulse on an iso vortical sheet by adapting this principle combined with an asymptotic solution of the euler equations we make an extension of fraenkel saffman s formula for the translation velocity of an axisymmetric vortex ring to third order in a small parameter the ratio of the core radius to the ring radius saffman s formula for a viscous vortex ring is also extended to third order

a special class in vanishing helicity ideal flows the derivation of a lagrangian invariant so called spirality is reviewed through the lagrangian coordinates the value of the spirality is fixed up to a gauge transformation the helicity conservation follows directly from this invariant among all ideal flows with zero helicity in a domain of flow frozen into the fluid motion a special class is introduced this special topological class has the possibility to transform its spirality to be identically zero everywhere in the domain for those simply connected domains of motion a necessary and a sufficient condition is presented for these zero spirality flows

magnetic fields and brownian motion on the 2 sphere using constrained path integrals the authors study some statistical properties of brownian paths on the two dimensional sphere a generalized levy law for the probability p a that a closed brownian path encloses an algebraic area a is obtained distributions of scaled variables related to the winding of paths around some fixed point are recovered in the asymptotic regime t to infinity

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

quasi monte carlo integration the standard monte carlo approach to evaluating multidimensional integrals using pseudo random integration nodes is frequently used when quadrature methods are too difficult or expensive to implement as an alternative to the random methods it has been suggested that lower error and improved convergence may be obtained by replacing the pseudo random sequences with more uniformly distributed sequences known as quasi random in this paper quasi random halton sobol and faure and pseudo random sequences are compared in computational experiments designed to determine the effects on convergence of certain properties of the integrand including variance variation smoothness and dimension the results show that variation which plays an important role in the theoretical upper bound given by the koksma hlawka inequality does not affect convergence while variance the determining factor in random monte carlo is shown to provide a rough upper bound but does not accurately predict performance in general quasi monte carlo methods are superior to random monte carlo but the advantage may be slight particularly in high dimensions or for integrands that are not smooth for discontinuous integrands we derive a bound which shows that the exponent for algebraic decay of the integration error from quasi monte carlo is only slightly larger than 1 2 in high dimensions more 21 refs 6 figs 5 tabs less

classifications of ideal 3d elastica shapes at equilibrium we investigate the equilibrium configurations of the ideal 3d elastica i e inextensible unshearable isotropic uniform and naturally straight and prismatic rods with linear elastic constitutive relations infinite solution trajectories are expressed analytically and classified in terms of only three parameters related to physical quantities orientation of sections and mechanical loading are also well described analytically with these parameters detailed analysis of solution trajectories yields two main results first all particular trajectories are completely characterized and located in the space of these parameters second a general geometric structure is exhibited for every ideal 3d elastic rod where the trajectory winds around a core helix in a tube shaped envelope this remarkable structure leads to a classification of the general case according to three properties called chirality components in addition the geometry of the envelope provides another characterization of the ideal 3d elas

the energy spectrum of a twisted flexible string under elastic relaxation a full investigation of the energy spectrum of a twisted flexible string under elastic relaxation is presented and discussed in detail for the first time new polynomial expressions for critical energy states are derived and the whole spectrum of critical states minima maxima and inflexion points of the elastic energy is found and discussed in relation to superhelicity and elastic characteristics of the string the study is carried out in the context of the theory of linear elasticity and the thin rod approximation the relaxation mechanism is studied by using conservation of linking difference by the formula delta lk wr delta tw we show how specific geometric quantities such as pitch angle writhe and twist contributions as well as physical quantities such as torsional and bending energy depend on superhelicity given by the specific linking difference and elastic characteristics of the string given by bending and torsional rigidity these quantities expressed per unit length are examined and compared at each critical energy state starting from a supertwisted configuration we show that the string relaxes by twist reduction through two different intermediate helical states which correspond to different local minima to reach the lowest minimum energy state in a supercoiled configuration the case of a generic kink formation and consequent passage through an inflexional configuration is then examined and new expressions for the energy change in the vicinity of the inflexion point are derived

generating quasi random paths for stochastic processes the need to simulate stochastic processes numerically arises in many fields frequently this is done by discretizing the process into small time steps and applying pseudorandom sequences to simulate the randomness this paper addresses the question of how to use quasi monte carlo methods to improve this simulation special techniques must be applied to avoid the problem of high dimensionality which arises when a large number of time steps is required two such techniques the generalized brownian bridge and particle reordering are described here these methods are applied to a problem from finance the valuation of a 30 year bond with monthly coupon payments assuming a mean reverting stochastic interest rate when expressed as an integral this problem is nominally 360 dimensional the analysis of the integrand presented here explains the effectiveness of the quasi random sequences on this high dimensional problem and suggests methods of variance reduction which can be used in conjunction with the quasi random sequences

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

conformal geometry of timelike curves in the 1 2 einstein universe abstract we study the conformal geometry of timelike curves in the 1 2 einstein universe the conformal compactification of minkowski 3 space defined as the quotient of the null cone of r 2 3 by the action by positive scalar multiplications the purpose is to describe local and global conformal invariants of timelike curves and to address the question of existence and properties of closed trajectories for the conformal strain functional some relations between the conformal geometry of timelike curves and the geometry of knots and links in the 3 sphere are discussed

particle simulations of the kac model of the boltzmann equation abstract we consider a simplified mathematical model of the boltzmann equation which was introduced by kac a quasi monte carlo particle simulation using t m s nets is described the particle movement is shown to be an evaluation of the volume of a subset of i4 0 1 4 an error bound for quasi monte carlo approximation of the volume of a generalized quadrant set is derived when a t m s net is used convergence of the simulation is proved when the particles are reordered according to their velocity in every time step quasi monte carlo and monte carlo particle simulations are compared in computational experiments the results indicate that quasi monte carlo simulation outperforms standard monte carlo approaches

knot complexity and related observables from path integrals for semiflexible polymers we analyse earlier obtained numerical results for the knot complexity and the topological persistence length i e the minimal number of steps on the lattice needed for the knot to be formed with help of well known path integrals for semiflexible chains in addition we provide an estimate for the number of different knots having knot diagrams with exactly n crossings analytical results obtained are in satisfactory agreement with the available numerical data

generation of ribbons helicoids and complex scherk surfaces in laser matter interactions motivated by a diversity of the shape instability phenomena in condensed matter physics we study the formation of elastic ribbon structures and transformation into helicoidal structures using the multipulse laser matter interaction with the co coated surface a one dimensional high density vortex filament array has been created increasing the number of pulses the oscillatory strain field causes the cascade of the shape transformations into structures of increasing topological complexity vortex filaments into ribbons into ribbon helicoids and tubular ribbon helicoids and then into short ribbon structures with the complex scherk surface being identified the cascade of transformations follows the scenario in which the topological complexity of the new structure increases with the number of pulses thus realizing configurations with more efficient energy relaxation above a critical number of pulses the system is catastrophically disintegrated into small scale wrinkled and crumpled surfaces we show that the critical parameter for the ribbon transformations is the number of laser pulses which is equivalent to some critical oscillatory frequency

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

multiple folding and packing in dna modeling in this paper we introduce and analyze a set of equations to study geometric and energetic aspects associated with the kinematics of multiple folding and coiling of closed filaments for dna modeling by these equations we demonstrate that a high degree of coiling may be achieved at relatively low energy costs through appropriate writhe and twist distribution and independently from the number of coils formed for sufficiently high twist we show that coiling is actually favoured by elastic energy relaxation when the deformation energy is due to curvature and the mean twist of the filament we give estimates for the writhing process based purely on curvature information and the number of coils produced we also determine the packing rate associated with filament compaction in the case of a hierarchical helical coiling these results find useful applications in dna biology especially in modeling dna wrapping in proteic regions where there is a strong connection between high coiling efficient compaction and energy localization

how tangled is a tangle abstract new measures of algebraic geometric and topological complexity are introduced and tested to quantify morphological aspects of a generic tangle of filaments the tangle is produced by standard numerical simulation of superfluid helium turbulence which we use as a benchmark for numerical investigation of complex systems we find that the measures used based on crossing number information are good indicators of generic behaviour and detect accurately a tangle s complexity direct measurements of kinetic helicity are found to be in agreement with the other complexity based measures proving that helicity is also a good indicator of structural complexity we find that complexity based measure growth rates are consistently similar to one another the growth rate of kinetic helicity is found to be twice that of energy

a discrete mechanics approach to the cosserat rod theory part 1 static equilibria a theory of discrete cosserat rods is formulated in the language of discrete lagrangian mechanics by exploiting kirchhoff s kinetic analogy the potential energy density of a rod is a function on the tangent bundle of the configuration manifold and thus formally corresponds to the lagrangian function of a dynamical system the equilibrium equations are derived from a variational principle using a formulation that involves null space matrices in this formulation no lagrange multipliers are necessary to enforce orthonormality of the directors noether s theorem relates first integrals of the equilibrium equations to lie group actions on the configuration bundle so called symmetries the symmetries relevant for rod mechanics are frame indifference isotropy and uniformity we show that a completely analogous and self contained theory of discrete rods can be formulated in which the arc length is a discrete variable ab initio in this formulation the potential energy density is defined directly on pairs of points along the arc length of the rod in analogy to veselov s discrete reformulation of lagrangian mechanics a discrete version of noether s theorem then identifies exact first integrals of the discrete equilibrium equations these exact conservation properties confer the discrete solutions accuracy and robustness as demonstrated by selected examples of application

writhe and mutual entanglement combine to give the entanglement length we propose a method to estimate n e the entanglement length that incorporates both local and global topological characteristics of chains in a melt under equilibrium conditions this estimate uses the writhe of the chains the writhe of the primitive paths and the number of kinks in the chains in a melt an advantage of this method is that it works for both linear and ring chains works under all periodic boundary conditions does not require knowing the contour length of the primitive paths and does not rely on a smooth set of data we apply this method to linear finitely extendable nonlinear elastic chains and we observe that our estimates are consistent with those from other studies

the energy and helicity of knotted magnetic flux tubes magnetic relaxation of a magnetic field embedded in a perfectly conducting incompressible fluid to minimum energy magnetostatic equilibrium states is considered it is supposed that the magnetic field is confined to a single flux tube which may be knotted a local non orthogonal coordinate system zero framed with respect to the knot is introduced and the field is decomposed into toroidal and poloidal ingredients with respect to this system the helicity of the field is then determined this vanishes for a field that is either purely toroidal or purely poloidal the magnetic energy functional is calculated under the simplifying assumptions that the tube is axially uniform and of circular cross section the case of a tube with helical axis is first considered and new results concerning kink mode instability and associated bifurcations are obtained the case of flux tubes in the form of torus knots is then considered and the ground state energy function m h where h is an internal twist parameter is obtained as expected m h which is a topological invariant of the knot increases with increasing knot complexity the function m h provides an upper bound on the corresponding function m h that applies when the above constraints on tube structure are removed the technique is applicable to any knot admitting a parametric representation on condition that points of vanishing curvature are excluded

a comparative study of pseudo and quasi random sequences for the solution intergral equations a numerical study for the solution of integral equations using random numbers from congruential generators and the halton and the faure sequences is presented the results indicate that for high precision monte carlo quasi random sequences yield better results in general the relative performance of the quasi random sequences is better with monte carlo schemes which reduce the dimensionality of the problem

the writhe of a self avoiding walk the writhe of a self avoiding walk in a three dimensional space is the average over all projections onto a plane of the sum of the signed crossings we compute this number using a monte carlo simulation our results suggest that the average of the absolute value of the writhe of self avoiding walks increases as nalpha where n is the length of the walks and alpha approximately 0 5 the mean crossing number of walks is also computed and found to have a power law dependence on the length of the walks in addition we consider the effects of solvent quality on the writhe and mean crossing number of walks

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

dynamics of helical strips the dynamics of inertial elastic helical thin rods with noncircular cross sections and arbitrary intrinsic curvature torsion and twist is studied the classical kirchhoff equations are used together with a perturbation scheme at the level of the director basis and the dispersion relation for helical strips is derived and analyzed it is shown that all naturally straight helical strips are unstable whereas free standing helices are always stable there exists a one parameter family of stationary helical solutions depending on the ratio of curvature to torsion a bifurcation analysis with respect to this parameter is performed and bifurcation curves in the space of elastic parameters are identified the different modes of instabilities are analyzed

geometrical equilibrium of curves a showcase of helical numerical solutions in this work we plot loop configurations that minimize energy functionals depending on geometrical invariants of the loop itself in particular we consider a family of functionals including curvature and torsion terms both linear and quadratic such that their combinations produce geometrical invariants in 1 noether s theorem was advantageously used to identify the constants of integration of the euler lagrange equations describing the equilibrium of loops for this family of energy functionals those authors demonstrated the integrability of these functionals by means of quadratures in this work we follow that approach to realize numeric calculations and to show plots of numerical solutions of the relevant equations for equilibrium of loops as these were presented but not studied in 1 we then show as the main result of this work a representative catalogue of such solutions in euclidean three dimensional space

the writhe of a self avoiding polygon we discuss the writhe of a self avoiding polygon on a lattice as a geometrical measure of its entanglement complexity we prove a rigorous result about the dependence of the absolute value of the writhe on the number n of edges in the polygon and use monte carlo methods to estimate the distribution of the writhe both for all polygons with n edges and for the subset of polygons that are trefoils

the statistical mechanics of a melt of polymer rings topological restrictions are introduced into a melt of flexible polymer rings by specifying the linking number between each pair of rings attention is focused on the configurational properties of a single ring in the melt where the winding number between each pair of rings is zero the configurations of all the other chains are averaged out in the partition sum the effect of the topological constraint on the remaining chain is expressed as a product of two configurational weighting factors which involve unusual geometrical properties of the configuration one is a phase factor depending on the torsion of the configuration and the other is a boltzmann like factor with the self inductance of the configuration in the role of the interaction energy the term based on the torsion has the remarkable property of transforming random walk like configurations into those of stiff rods while the inductance term promotes a transition to a completely collapsed state our preliminary results suggest that the actual configuration of the loop is a non trivial balance between these opposing tendencies with the size r of the loop scaling with the length as r2 approximately l1 1 3 pi approximately l0 89

solving the boundary value problem for finite kirchhoff rods the kirchhoff model describes the statics and dynamics of thin rods within the approximations of the linear elasticity theory in this paper we develop a method based on a shooting technique to find equilibrium configurations of finite rods subjected to boundary conditions and given load parameters the method consists in making a series of small changes on a trial solution satisfying the kirchhoff equations but not necessarily the boundary conditions by linearizing the differential equations around the trial solution we are able to push its end point to the desired position step by step the method is also useful to obtain configurations of rods with fixed end points but different mechanical parameters such as tension components of the moment or inhomogeneities

topological invariants in period doubling cascades topological characterization is a useful description of dynamical behaviours as exemplified by templates which synthesize the topological properties of very dissipative chaotic attractors embedded in tri dimensional phase spaces such a description relies on topological invariants such as linking numbers between two periodic orbits which may be viewed as knots these invariants may therefore be used to understand the structure of dynamical behaviours nevertheless as an example the celebrated period doubling cascade is usually investigated by using total twists which are not topological invariants instead we introduce linking numbers between an orbit viewed as the core of a small ribbon and the edges of the ribbon such a linking number which is in fact the calugareanu invariant is related to the total twist number and the number of writhes of the ribbon a second topological invariant called the effective twist number is also introduced and is useful for investigating period doubling cascades in the case of a trivial suspension of a horseshoe map this topological invariant may be predicted from a symbolic dynamics with the aid of framed braid representations