a cellular automaton model of excitable media iii fitting the belousov zhabotinskiic reaction abstract we obtain a set of parameter values for our cellular automaton model that gives a reasonably good description of the belousov zhabotinskii reaction as determined by comparison with experimental results and with calculations on the oregonator a differential equation model of the chemical system

parametric forcing of scroll wave patterns in three dimensional excitable media abstract the dynamics of axisymmetric and twisted scroll rings under homogeneous periodic forcing is studied numerically the collapse of axisymmetric rings can be enhanced or retarded significantly with resonant forcing twisted scroll rings exhibit the phenomenon of resonant drift in a direction normal to the axis of the central filament the scaling of mean and fluctuating axial drift due to forcing is determined evidence is provided showing that the appropriate symmetry group for the dynamics of the unforced twisted scroll ring is se 2 r ordinary differential equations based on symmetry considerations explain the dynamics without resorting to laws of filament motion and the local geometry hypothesis

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

microlocal analysis of some isospectral deformations we study the microlocal structure of the examples of isospectral deformations of riemannian manifolds given by d deturck and c gordon in det gi the schwartz kernel of the intertwining operators considered by them may be written as an oscillatory integral with a singular phase function and product type amplitude in certain instances we identify them as belonging to the space of fourier integral operators associated with various pairwise intersecting lagrangians after formulating a class of operators incorporating the most relevant features of the operators above we establish a composition calculus for this class and show that is not necessary to introduce new lagrangians in the composition

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

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

a cellular automaton model of excitable media ii curvature dispersion rotating waves and meandering waves abstract a cellular automaton model of excitable media recently introduced by the authors is examined in more detail here we show that i the model is consistent with a linear dependence of wave speed on wave front curvature in two dimensional space ii the model exhibits a stepwise increasing relation between wave speed and temporal period the dispersion relation whose precise shape can be controlled by parameter settings iii the model generates rigidly rotating spiral waves whose frequency speed and core size are consistent with theoretical expectations and iv spiral waves rotate non rigidly showing short term flowerlet meandering which seems to arise by bifurcation from the rigidly rotating solution and long term irregular wandering over large distances these observations are consistent with many known features of wave propagation in two dimensional continuous excitable media governed by reaction diffusion equations suggesting that our cellular automaton model is not only a fast but also a reliable method for investigating the behavior of excitable media

stabilization of a scroll ring by a cylindrical neumann boundary we study the interaction of phase singularities with homogeneous neumann boundaries in one two and three spatial dimensions for the complex ginzburg landau equation the existence of a boundary induced drift attractor well known for spiral waves in two spatial dimensions is demonstrated for scroll waves in three spatial dimensions we find that a cylindrical neumann boundary can lock a scroll ring thus preventing the collapse of its closed filament

isospectral deformations i riemannian structures on two step nilspaces on montre comment construire des exemples de deformations isospectrales de metriques de riemann sur des varietes sur lesquelles des recouvrements de certains groupes de lie nilpotents agissent

a geometric theory for scroll wave filaments in anisotropic excitable media scroll waves are an important example of self organisation in excitable media in cardiac tissue scroll waves of electrical activity underlie lethal ventricular arrhythmias and fibrillation they rotate around a topological line defect which has been termed the filament numerical investigation has shown that anisotropy can substantially affect the dynamics of scroll waves it has recently been hypothesised that stationary scroll wave filaments in cardiac tissue describe geodesics in a space whose metric is the inverse diffusion tensor several computational studies have validated this hypothesis but until now no quantitative theory has been provided to study the effects of anisotropy on scroll wave filaments here we review in detail the recently developed covariant formalism for scroll wave dynamics in general anisotropy and derive the equations of motion of filaments these equations are fully covariant under general spatial coordinate transformations and describe the motion of filaments in a curved space whose metric tensor is the inverse diffusion tensor our dynamic equations are valid for thin filaments and for general anisotropy and we show that stationary filaments obey the geodesic equation we extend previous work by allowing spatial variations in the determinant of the diffusion tensor and the reaction parameters leading to drift of the filament

remarks on parameter identification ii convergence of the algorithm on demontre un theoreme de convergence pour l algorithme d identification du coefficient a x de l equation parabolique u t a x u x x f x t et on donne des resultats de stabilite

calcul du spectre d une nilvariete de rang deux et applications on calcule en utilisant la theorie des orbites de kirillov le spectre d une nilvariete compacte de rang deux puis on utilise ce calcul pour etudier et caracteriser les deformations isospectrales de ces varietes

on isospectral periodic potentials on a discrete lattice i on considere une version discretisee et generalisee de l equation de schrodinger et on etudie le probleme aux valeurs propres

equivalence of quasi regular representations of two and three step nilpotent lie groups abstract we characterize all pairs of cocompact discrete subgroups 1 and 2 of a two step nilpotent lie group m such that the quasi regular representations of m on l 2 1 m and l 2 2 m are unitarily equivalent this characterization is used to construct new examples we also describe a pair of nonisomorphic cocompact discrete subgroups 1 and 2 of a three step nilpotent lie group m such that the quasi regular representations of m on l 2 1 m and l 2 2 m are unitarily equivalent

line bundle laplacians over isospectral nilmanifolds we show that nontrivial isospectral deformations of a big class of compact riemannian two step nilmanifolds can be distinguished from trivial deformations by the behaviour of bundle laplacians on certain non flat hermitian line bundles over these manifolds

geometry of 2 step nilpotent groups with a left invariant metric ii we obtain a partial description of the totally geodesic submanifolds of a 2 step simply connected nilpotent lie group with a left invariant metric we consider only the case that n is nonsingular that is ad x y is surjective for all elements 4 e x y where x denotes the lie algebra of n and y denotes the center of x among other results we show that if h is a totally geodesic submanifold of n with dimh 1 dimy then h is an open subset of gn where g is an element of h and n is a totally geodesic subgroup of n we find simple and useful criteria that are necessary and sufficient for a subalgebra x of x to be the lie algebra of a totally geodesic subgroup n we define and study the properties of a gauss map of a totally geodesic submanifold h of n we conclude with a characterization of 2 step nilpotent lie groups n of heisenberg type in terms of the abundance of totally geodesic submanifolds of n

the dynamics of scroll waves in excitable media an overview is given of the attempts to understand the behavior of scroll waves in three dimensional excitable media using analytical methods and these results are compared with recent numerical and experimental results the behavior of untwisted planar scroll rings twisted rings and helices and knotted scroll wave filaments is discussed

isospectral metrics on five dimensional spheres we construct isospectral pairs of riemannian metrics on s 5 and on b 6 thus lowering by three the minimal dimension of spheres and balls on which such metrics have been constructed previously s n 8 and b n 9 we also construct continuous families of isospectral riemannian metrics on s 7 and on b 8 in each of these examples the metrics can be chosen equal to the standard metric outside certain subsets of arbitrarily small volume

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

h type groups and iwasawa decompositions since their introduction by a kaplan kpl some ten years ago generalised heisenberg groups also known as groups of heisenberg type or h type groups have provided a framework in which to construct interesting examples in geometry and analysis see for instance c2 kp2 kp3 kpr k2 rl r2 tl tv the iwasawa n groups associated to all the real rank one simple lie groups are h type so one has a convenient vehicle for studying these in a unified way many problems on these simple lie groups can be reduced to a problem on h type groups via the so called noncompact picture and often problems on h type groups can be solved on all the groups of the family in one fell swoop as in for example ch ck dr out of this approach to studying simple lie groups several problems arise such as why only some h type groups correspond to simple lie groups of real rank one in this paper we discuss various features of iwasawa n groups which distinguish them in the class of all h type groups we shall show that all h type groups which possess certain geometric properties clearly possessed by iwasawa n groups satisfy a lie algebraic condition implicit in the work of b kostant kt2 that we shall call the j condition we shall also use elementary clifford algebra to classify the

computation of the response functions of spiral waves in active media rotating spiral waves are a form of self organization observed in spatially extended systems of physical chemical and biological natures a small perturbation causes gradual change in spatial location of spiral s rotation center and frequency i e drift the response functions rfs of a spiral wave are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues lambda 0 i omega the rfs describe the spiral s sensitivity to small perturbations in the way that a spiral is insensitive to small perturbations where its rfs are close to zero the velocity of a spiral s drift is proportional to the convolution of rfs with the perturbation here we develop a regular and generic method of computing the rfs of stationary rotating spirals in reaction diffusion equations we demonstrate the method on the fitzhugh nagumo system and also show convergence of the method with respect to the computational parameters i e discretization steps and size of the medium the obtained rfs are localized at the spiral s core

large excitability asymptotics for scroll waves in three dimensional excitable media three dimensional scroll waves are considered in a reaction diffusion model of excitable media in the large excitability limit coordinates based on the scroll filament are defined and shown to provide a natural extension of the coordinates used for two dimensional spiral waves the leading order free boundary equations for interface motion in three dimensions are explicitly derived in these coordinates three specific examples are considered straight twisted scroll waves axisymmetric scroll waves and helical scroll waves the equations for the fields at leading and first order in the core region are given

singular filaments organize chemical waves in three dimensions iii knotted waves abstract this is the third of a series of papers on the anatomy of three dimensional organizing centers in excitable media we here ask whether all self consistent waves in excitable media are topologically equivalent to the experimentally verified scroll ring whose axis lies in a plane as a test case we examine a scroll ring whose axis contains a knot it proves to be incompatible with the requirements of physical chemistry unless simultaneously twisted by an amount equal to the writhing number of its axis which is zero for planar closed curves appropriate initial conditions are suggested for experimentally creating a wave whose source is a scroll ring knotted and twisted in this way

stationary self propagating fronts in potential flow abstract we analyze the problem of stationary self propagating fronts in potential flow the issues of local existence and uniqueness for solutions of the ode describing stationary fronts multiplicity of solutions and linearized stability of a stationary front as a solution of the hyperbolic evolution equation are addressed the results are illustrated in the case of the dual source system which is a simple model of a combustion system in which local extinction may arise model extensions for combustion applications are presented

remarks on parameter identification i in this paper a method for constructing a spatially varying diffusion coefficient for a parabolic partial differential equation is given this function is obtained as the limit of a sequence of functions which are obtained by solving a sequence of finite dimensional optimization problems

helical and circular scroll wave filaments abstract we present a formal theory of the motion of circular and helical scroll wave filaments with uniform twist some implications of the theory are derived and compared to numerical computations of scroll wave evolution

integrability of geodesic flows and isospectrality of riemannian manifolds we construct a pair of compact eight dimensional two step riemannian nilmanifolds m and m which are isospectral for the laplace operator on functions and such that m has completely integrable geodesic flow in the sense of liouville while m has not moreover for both manifolds we analyze the structure of the submanifolds of the unit tangent bundle given by two maximal continuous families of closed geodesics with generic velocity fields the structure of these submanifolds turns out to reflect the above non integrability properties on the other hand their dimension is larger than that of the lagrangian tori in m indicating a degeneracy which might explain the fact that the wave invariants do not distinguish an integrable from a nonintegrable system here finally we show that for m the invariant eight dimensional tori which are foliated by closed geodesics are dense in the unit tangent bundle and that both m and m satisfy the so called clean intersection hypothesis

continuous families of isospectral metrics on simply connected manifolds we construct continuous families of riemannian metrics on certain simply connected manifolds with the property that the resulting riemannian manifolds are pairwise isospectral for the laplace operator acting on functions these are the first examples of simply connected riemannian manifolds without boundary which are isospectral but not isometric for example we construct continuous isospectral families of metrics on the product of spheres s4 x s3 x s3 the metrics considered are not locally homogeneous for a big class of such families the set of critical values of the scalar curvature function changes during the deformation moreover the manifolds are in general not isospectral for the laplace operator acting on 1 forms

spiral wave solutions for reaction diffusion equations in a fast reaction slow diffusion limit abstract a two species reaction diffusion equation is studied in a distinguished limit of fast reaction and slow diffusion asymptotic matching yields a spiral solution whose rotation frequency is a unique function of the physical parameters and the constant pitch of the spiral using an idea of fife the problem is first reduced to a free boundary problem the solution is obtained by matching the core where diffusion balances the curvature of the free boundary to the far field where the effects of diffusion are small the equations of motion are averaged over the rotation frequency to yield steady equations in the two regions helically symmetric solutions to the averaged equation reduce to solving a fourth order ode in the core region matched to a second order ode in the far field speculations on the dynamics of the system and coupling between the core and far field are presented

the structure of underlying lie algebras abstract every lie algebra over a field e gives rise to new lie algebras over any subfield f e by restricting the scalar multiplication this paper studies the structure of these underlying lie algebra in relation to the structure of the original lie algebra in particular the question how much of the original lie algebra can be recovered from its underlying lie algebra over subfields f by introducing the conjugate of a lie algebra we show that in some specific cases the lie algebra is completely determined by its underlying lie algebra furthermore we construct examples showing that these assumptions are necessary as an application we give for every positive n an example of a real 2 step nilpotent lie algebra which has exactly n different bi invariant complex structures this answers an open question by di scala lauret and vezzoni motivated by their work on quasi kahler chern flat manifolds in differential geometry the methods we develop work for general lie algebras and for general galois extensions f e in contrast to the original question which only considered nilpotent lie algebras of nilpotency class 2 and the field extension r c we demonstrate this increased generality by characterizing the complex lie algebras of dimension 4 which are defined over r and over q

a cellular automation model of excitable media iv untwisted scroll rings abstract we study the motion of untwisted circular scroll rings in an excitable medium modeled by a cellular automation that includes the effects of curvature and dispersion on wave propagation we find that scroll rings either shrink or expand with a velocity that is a function of the curvature of the ring and they drift along their axis of symmetry

a cornucopia of isospectral pairs of metrics on spheres with different local geometries this article concludes the comprehensive study started in sz5 where the first nontrivial isospectral pairs of metrics are constructed on balls and spheres these investigations incorporate four different cases since these balls and spheres are considered both on 2 step nilpotent lie groups and on their solvable extensions in sz5 the considerations are completely concluded in the ball case and in the nilpotent case the other cases were mostly outlined in this paper the isospectrality theorems are completely established on spheres also the important details required about the solvable extensions are concluded in this paper

dynamics of dissipative structures in reaction diffusion equations the authors investigate the dynamics of dissipative structures in a reaction diffusion system they propose an analytical theory for the behavior of dissipative structures and show that a dissipative structure ds that is stable in a one dimensional homogeneous medium can be induced to drift by slow variation of the diffusion coefficients or by curvature of the ds in higher dimensions in one spatial dimension this motion can be in the direction of increasing or decreasing diffusion coefficient depending on properties of the ds which can be determined analytically in two and three dimensions this drift is proportional to the sum of the curvature of the ds and the gradient of the diffusion coefficient of the medium the analysis of this motion uses standard ideas from perturbation theory to find an equation of motion for the location of the ds numerical simulations in one and two dimensions show good quantitative agreement with the theoretical results

riemannian nilmanifolds and the trace formula this paper examines the clean intersection hypothesis required for the expression of the wave invariants computed from the asymptotic expansion of the classical wave trace by duistermaat and guillemin the main result of this paper is the calculation of a necessary and sufficient condition for an arbitrary riemannian two step nilmanifold to satisfy the clean intersection hypothesis this condition is stated in terms of metric lie algebra data we use the calculation to show that generic two step nilmanifolds satisfy the clean intersection hypothesis in contrast we also show that the family of two step nilmanifolds that fail the clean intersection hypothesis are dense in the family of two step nilmanifolds finally we give examples of nilmanifolds that fail the clean intersection hypothesis

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

dynamics of wave fronts in excitable media abstract the dynamics of wave fronts in two dimensional excitable media is analytically studied by means of the asymptotic perturbation method for wave fronts in reaction diffusion systems we derive a space derivative burgers equation which governs the shape of boundary layers and their propagation application of lie group analysis of differential equations yields group invariant solutions of this equation and their classification some classes of the group invariant solutions determine the remarkable patterns expanding circle waves rotating spiral waves initial value problems for the burgers equation are also studied and a spiral wave is shown to be asymptotically realized under some proper boundary conditions

isospectral deformations of metrics on spheres we construct non trivial continuous isospectral deformations of riemannian metrics on the ball and on the sphere in r n for every n 9 the metrics on the sphere can be chosen arbitrarily close to the round metric in particular they can be chosen to be positively curved the metrics on the ball are both dirichlet and neumann isospectral and can be chosen arbitrarily close to the flat metric

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

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

identification of parameters in distributed parameter systems by regularization identification of spatially varying parameters in distributed parameter systems from noisy data is an ill posed problem the concept of regularization widely used in solving linear fredholm integral equations is developed for the identification of parameters in distributed parameter systems a general regularization identification theory is first presented and then applied to the identification of parabolic systems the performance of the regularization identification method is evaluated by numerical experiments on the identification of a spatially varying diffusivity in the diffusion equation

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

singular perturbation theory of traveling waves in excitable media a review abstract waves of chemical or electrical activity traveling through space have been observed in several contexts chemical reaction mixtures in non convecting liquid phase cell suspensions nerve axons and neuromuscular tissues typically wave supporting preparations are excitable that is they respond and sensitivity to perturbations are rapidly damped out but suprathreshold disturbances trigger an abrupt and substantial response the abruptness of the response can be exploited by the methods of singular perturbation theory to obtain a mathematical description of wave propagation in spatially distributed excitable media singular perturbation analysis of propagating waves in one spatial dimension is straightforward and uncontentious but the analysis of propagating waves in one spatial dimension is straightforward and several fundamentally different ways we compare and contrast the approaches taken by greenberg zykov fife krinskii and his collaborators and ourselves with particular emphasis on the case of rotating spiral waves our intention is to bring some order to the important but difficult theory of propagating waves in two dimensional excitable media in conclusion we discuss briefly some possible extensions of the singular perturbation approach to propagating wave surfaces in three dimensional space

isospectral riemannian metrics and potentials pour une variete de riemann compacte m g on considere le spectre de l operateur de laplace beltrami et de l operateur de schrodinger v agissant sur l u 2 m g

zero entropy non integrable geodesic flows and a non commutative rotation vector let g be a 2 step nilpotent lie algebra we say g is non integrable if for a generic pair of points p p g the isotropy algebras do not commute g p g p 0 theorem if g is a simply connected 2 step nilpotent lie group g lie g is non integrable d g is a cocompact subgroup and g is a left invariant riemannian metric then the geodesic flow of g on t d g is neither liouville nor non commutatively integrable with c first integrals the proof uses a generalization of the rotation vector pioneered by benardete and mitchell