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

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

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

pattern synthesis from singular solutions in the debye limit helical waves and twisted toroidal scroll structures applying the debye limit to a suitable superposition of singular solutions of the linear two component reaction diffusion equations in three dimensions leads to helical and twisted toroidal scroll waves of constant concentration an integral representation of the solutions of the helmholtz equation in toroidal coordinates establishes a connection between helical and toroidal waves

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

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

a geometrical theory for spiral waves in excitable media in this paper we develop a geometrical theory for waves in excitable reacting media using singular perturbation arguments and dispersion of traveling plane wave trains we derive an approximate theory of wave front propagation which has strong resemblance to the geometrical diffraction theory of high frequency waves in hyperbolic systems governed by the eikonal equation using this theory we study the effect of curvature on waves in excitable media specifically rotating spiral patterns in planar regions from this theory we are able to determine the frequency and wavelength for spiral patterns in excitable nonoscillatory media

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

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

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

towards efficient tracking of inertial particles with high order multidomain methods this paper develops an efficient particle tracking algorithm to be used in fluid simulations approximated by a high order multidomain discretization of the navier stokes equations we discuss how to locate a particle s host subdomain how to interpolate the flow field to its location and how to integrate its motion in time a search algorithm for the nearest subdomain and quadrature point tuned to a typical quadrilateral isoparametric spectral subdomain takes advantage of the inverse of the linear blending equation we show that to compute particle laden flows a sixth order lagrangian polynomial that uses points solely within a subdomain is sufficiently accurate to interpolate the carrier phase variables to the particle position time integration of particles with a lower order adams bashforth scheme rather than the fourth order runge kutta scheme often used for the integration of the carrier phase increases computational efficiency while maintaining engineering accuracy we verify the tracking algorithm with numerical tests on a steady channel flow and an unsteady backward facing step flow

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

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

spiral waves in the belousov zhabotinskii reaction abstract the beautiful spiral waves of oxidation in the belousov zhabotinskii reaction are the source of many interesting and important questions about periods structures in excitable media it has long been known that these spirals are similar to involutes of circles at least some distance from the center but until now no way has been known to determine the correct wavelength and frequency in this paper we show that the parameters of a spiral wave can be viwed s eigenvalues of a problem with unique solution the critical ingredients of the theory are the effects of curvature on the propagation of wavefronts in two dimensional media and the dispersion of plane waves our analytical results are shown to be in good agreement with experimental data for the belousov zhabotinskii reagent

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

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

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

singular filaments organize chemical waves in three dimensions iv wave taxonomy abstract this is the last of four papers on the anatomy of three dimensional organizing centers in excitable media these organizing centers are made of scroll ring waves with twists and knots in them diversely linked together each such aggregate of linked rings is characterized by a handful of topological quantum numbers which partition the family of organizing centers into discrete classes or atoms we derive the rules which govern nuclear structure in this periodic table as well as the selection rules for transmutation

flip a method for adaptively zoned particle in cell calculations of fluid flows in two dimensions a method is presented for calculating fluid flow in two dimensions using a full particle in cell representation on an adaptively zoned grid the method has many interesting properties among them an almost total absence of numerical dissipation and the ability to represent large variations in the data the method is described using a standard formalism and its properties are illustrated by supersonic flow over a step and the interaction of a shock with a thin foil

a finite element semi lagrangian method with l2 interpolation summary high order accurate methods for convection dominated problems have the potential to reduce the computational effort required for a given order of solution accuracy the state of the art in this field is more advanced for eulerian methods than for semi lagrangian slag methods in this paper we introduce a new slag method that is based on combining the modified method of characteristics with a high order interpolating procedure the method employs the finite element method on triangular meshes for the spatial discretization an l2 interpolation procedure is developed by tracking the feet of the characteristic lines from the integration nodes numerical results are illustrated for a linear advection diffusion equation with known analytical solution and for the viscous burgers equation the computed results support our expectations for a robust and highly accurate finite element slag method copyright 2012 john wiley sons ltd

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

algebraic geometric codes and multidimensional cyclic codes a unified theory and algorithms for decoding using grobner bases it is proved that any algebraic geometric ag code can be expressed as a cross section of an extended multidimensional cyclic code both ag codes and multidimensional cyclic codes are described by a unified theory of linear block codes defined over point sets ag codes are defined over the points of an algebraic curve and an m dimensional cyclic code is defined over the points in m dimensional space the power of the unified theory is in its description of decoding techniques using grobner bases in order to fit an ag code into this theory a change of coordinates must be applied to the curve over which the code is defined so that the curve is in special position for curves in special position all computations can be performed with polynomials and this also makes it possible to use the theory of grobner bases next a transform is defined for ag codes which generalizes the discrete fourier transform the transform is also related to a grobner basis and is useful in setting up the decoding problem in the decoding problem a key step is finding a grobner basis for an error locator ideal for ag codes multidimensional cyclic codes and indeed any cross section of an extended multidimensional cyclic code sakata s algorithm can be used to find linear recursion relations which hold on the syndrome array in this general context the authors give a self contained and simplified presentation of sakata s algorithm and present a general framework for decoding algorithms for this family of codes in which the use of sakata s algorithm is supplemented by a procedure for extending the syndrome array

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

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

an anisotropic fully adaptive algorithm for the solution of convection dominated equations with semi lagrangian schemes abstract we present in this paper an anisotropic fully adaptive spatial temporal algorithm for the solution of convection dominated equations in the context of semi lagrangian schemes we devise the algorithm within a finite element framework suitable for higher order finite elements and derive a newly proposed a posteriori error indicator which allows us to control the local or truncation error in the l 2 norm at each time step this a posteriori error is split into temporal and spatial contributions leading us to define an optimal time step size and an optimal triangulation respectively as regards the spatial adaptation anisotropic unstructured triangular meshes are used to capture the distinctive features of the evolving discrete solution of the governing equations for solutions exhibiting strong anisotropies the orientation shape and size of the mesh triangles are provided by a metric tensor valid for linear and quadratic finite elements finally we show the capabilities of the algorithm for linear and quadratic finite elements by a series of two and three dimensional benchmarks taken from the literature involving purely convective as well as convection dominated problems

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

application of bubnov galerkin formulation to orthogonal grid generation abstract a bubnov galerkin formulation is used to solve an elliptic grid generation system by using linear and quadratic isoparametric elements good orthogonality characteristics are obtained for symmetric and non symmetric physical domains using both complete boundary correspondence or a combination of dirichlet and neuman boundary conditions the method exhibits excellent stability and requires a low number of iterations to attain convergence results are compared with those presented in e d chikliwala and y c yortsos j comput phys 57 391 1985

a nearly conservative high order lagrange galerkin method for the resolution of scalar convection dominated equations in non divergence free velocity fields abstract in this work we present a novel lagrange galerkin method for the resolution of the scalar pure convection and convection dominated diffusion equations in non divergence free velocity fields the scheme has been formulated so that i time integration is carried out by means of an arbitrary order backward differentiation formula ii finite element space discretizations of any order can be employed and iii it satisfies a discrete mass balance equation at each simulation instant and preserves mass in those cases in which no fluid enters nor leaves the control domain as long as the integrals involved are computed exactly these properties are achieved by posing a conservation law for a weighted mass from which the standard mass can be seen as a particular case that can be easily discretized in time and in space with any order of accuracy numerical experiments which include a three dimensional test show that the method is as accurate in time as the employed time marching formula as accurate in space as the finite element discretization and generally more accurate especially in terms of mass conservation errors and efficient than the classic non conservative lagrange galerkin method to the best of our knowledge this is the first conservative lagrange galerkin method for convection dominated equations that considers both non divergence free velocity fields and time and space discretizations of any order

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

an integral invariant for scroll rings in a reaction diffusion system abstract the evolution of scroll rings with filaments of arbitrary shape has been studied analytically for reaction diffusion systems with all the state variables having the same diffusion coefficients belonging to this class of systems are in particular models of the belousov zhabotinskii bz chemical reaction it has been found that the speed of decrease of an area bounded by the planar scroll filament is given by s 2 d d is the diffusion coefficient and is independent of the kind of model used to describe an active medium time and filament shape i e the integral d s 2 d where d s is the differential of the filament length and is the drift velocity always remains valid using this integral invariant a number of problems on the drift of closed and unclosed three dimensional vortices have been solved the theoretical predictions have been verified in experiments with the bz reaction

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

syndrome generation and error location search for the decoding of algebraic geometry codes on plane garcia stichtenoth curves in this paper the basic function field of an asymptotically optimal tower of function fields by garcia and stichtenoth is studied where bases of one point linear systems can be explicitly constructed a representation of finite gf q 2 rational points on the associated plane curve is derived in this paper this representation is exploited to extend the use of horner s loops and the mechanism of chien search in the decoding of reed solomon codes for syndrome generation and error location search respectively to algebraic geometry codes built on the plane garcia stichtenoth curve

a local anisotropic adaptive algorithm for the solution of low mach transient combustion problems a novel numerical algorithm for the simulation of transient combustion problems at low mach and moderately high reynolds numbers is presented these problems are often characterized by the existence of a large disparity of length and time scales resulting in the development of directional flow features such as slender jets boundary layers mixing layers or flame fronts this makes local anisotropic adaptive techniques quite advantageous computationally in this work we propose a local anisotropic refinement algorithm using for the spatial discretization unstructured triangular elements in a finite element framework for the time integration the problem is formulated in the context of semi lagrangian schemes introducing the semi lagrange galerkin slg technique as a better alternative to the classical semi lagrangian sl interpolation the good performance of the numerical algorithm is illustrated by solving a canonical laminar combustion problem the flame vortex interaction first a premixed methane air flame vortex interaction with simplified transport and chemistry description test i is considered results are found to be in excellent agreement with those in the literature proving the superior performance of the slg scheme when compared with the classical sl technique and the advantage of using anisotropic adaptation instead of uniform meshes or isotropic mesh refinement as a more realistic example we then conduct simulations of non premixed hydrogen air flame vortex interactions test ii using a more complex combustion model which involves state of the art transport and chemical kinetics in addition to the analysis of the numerical features this second example allows us to perform a satisfactory comparison with experimental visualizations taken from the literature

a generalized particle search locate algorithm for arbitrary grids a generalized iterative algorithm for searching and locating particles in arbitrary meshes is presented the algorithm uses newton method to invert a bijective map of the mesh elements onto a reference element together with a criterion to move from element to element in the mesh the generality of the method is shown by explicit formulations for linear and quadratic triangular and quadrilateral elements numerical examples demonstrate the performance of the method as well as its higher accuracy and versatility

architectures for syndrome generation and error location search in the decoding of hermitian codes when a hermitian curve is put in a special position with respect to its infinite rational point the x y coordinates of all its finite rational points have regular algebraic properties based on these properties the use of horner s rule and the mechanism of chien search in the decoding of reed solomon codes can be extended to render architectures for syndrome generation and error location search in the decoding of codes constructed from the hermitian curve

a galerkin characteristic algorithm for transport diffusion equations we introduce in this paper an algorithm for integration of the transport diffusion equations by combining the galerkin finite element method with the discretization of the total derivative along th

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

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

modified lagrange galerkin methods of first and second order in time for convection diffusion problems we introduce modified lagrange galerkin mlg methods of order one and two with respect to time to integrate convection diffusion equations as numerical tests show the new methods are more efficient but maintaining the same order of convergence than the conventional lagrange galerkin lg methods when they are used with either p 1 or p 2 finite elements the error analysis reveals that 1 when the problem is diffusion dominated the convergence of the modified lg methods is of the form o h m 1 h 2 t q q 1 or 2 and m being the degree of the polynomials of the finite elements 2 when the problem is convection dominated and the time step t is large enough the convergence is of the form o frac h m 1 delta t h 2 delta t q 3 as in case 2 but with t small then the order of convergence is now o h m h 2 t q 4 when the problem is convection dominated the convergence is uniform with respect to the diffusion parameter x t so that when 0 and the forcing term is also equal to zero the error tends to that of the pure convection problem our error analysis shows that the conventional lg methods exhibit the same error behavior as the mlg methods but without the term h 2 numerical experiments support these theoretical results

the dynamics of three dimensional scroll waves in excitable media abstract we investigate the motion of the central filament of a scroll wave in excitable medium we assume that two dimensional spiral solutions are known and use these solutions to produce slowly varying scroll waves in three dimensional space using standard ideas from perturbation theory multiscale techniques we find equations governing the motion of the central filament of the scroll and the twist of the scroll wave these equations are specialized to the case of a circular filament with uniformly distributed twist to determine the collapse or expansion and vertical drift of the spiral filament and to a helical filament with drift the predictions of these equations are in good qualitative agreement with the numerical experiments

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

decoding algebraic geometric codes up to the designed minimum distance a simple decoding procedure for algebraic geometric codes c sub omega d g is presented this decoding procedure is a generalization of peterson s decoding procedure for the bch codes it can be used to correct any d 1 2 or fewer errors with complexity o n sup 3 where d is the designed minimum distance of the algebraic geometric code and n is the codelength

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

knotted scroll wave filaments in excitable media abstract it is shown that certain general systems of diffusion reaction equations can have invariant knotted scroll wave solutions the theory is based on the assumption that the system of equations supports permanently rotating spiral solutions and from this assumptions equations governing the motion of the scroll wave filament can be derived then bifurcatilon theory is used to show that the equations of motion have torus knots as invariant solutions analytical methods are used to determine qualitative and quantitative features of these knotted solutions

wavefront geometry in excitable media abstract in a certain kind of continuous medium an isolated fragment of wavefront behaves peculiarly its edge a closed ring remains stationary while the fragment s interior continues its forward propagation curling into a scroll like shape along the edge a scroll ring the edge the axis of the scroll ring is a 1 dimensional singularity in the resulting periodic wave field from numerical simulations in 2 dimensional continua and from laboratory experiments in 3 dimensional continua this seems to be typical behavior for so called excitable media the implications include a bestiary of organizing centers each composed of a topologically distinct arrangement of linked and knotted scroll rings two examples are given exact description of these organizing centers awaits formulation of a tractable equation that embodies the essential features of an excitable medium

dynamical stability of untwisted scroll rings in excitable media abstract we report numerical simulations of the dynamical behavior of three dimensional untwisted scroll rings for three different models of excitable media the computations reveal for the first time the existence of a stable steadily rotating non drifting as well as non expanding shrinking scroll wave near a wall expansion of the scroll ring is observed for different choices of kinetic models along with contraction and drift numerical calculations also indicate aperiodic or meandering movement of the three dimensional scroll ring near the wall contrary to earlier observations some scroll rings do not drift noticeably at any value of the diffusion coefficient of the recovery variable

stable three dimensional action potential circulation in the fitzhugh nagumo model abstract excitable media generally support vortex rings of self excitation depending on the exact nature of the medium such a ring may expand or contract possibly to a stable radius we describe one such case encountered during numerical experiments on a simple model of electrophysiological excitability in nerve and cardiac muscle membrane the ring s rate of shrinkage depends parabolically on curvature and on proximity to other rings the vortex period also depends on curvature so rings of different sizes compete for territory we associate the stability of the ring with repulsive forces which we show are present between two dimensional rotors the observed minimal distance for repulsion agrees with the stable radius of the vortex ring

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

a new approach for conjugate heat transfer problems using immersed boundary method for curvilinear grid based solvers abstract use of immersed boundary method ibm based techniques have helped considerably in easing the grid generation process in flows involving complex geometries and or large boundary movements body fitting grid based techniques still however are advantageous in terms of accuracy and efficiency in this work we have developed an ibm scheme applicable to curvilinear coordinates aiming at taking advantage of both the methodologies the framework uses efficient algorithms for search locate and interpolate operations a new method of implementing the conjugate heat transfer cht boundary condition is proposed which is a direct extension of the method used for other boundary conditions and does not involve any complex interpolations like previous cht implementations using ibm the developed scheme is shown to be applicable to complex geometries on curvilinear grids while also being very efficient consuming less than 1 of the total simulation time per time step very good scalability on massive computations is demonstrated using strong scaling study up to 1024 cores detailed code verification process is undertaken to show that the method is second order accurate for both the velocity and temperature fields for all the boundary conditions considered further validation studies involving uniform flow over stationary and oscillating cylinders are carried out to demonstrate the accuracy of the developed method lastly simulations are performed to study flow and conjugate heat transfer through thick walled micro channels using body fitted background grids and the results are shown to be in excellent agreement with previously published results