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

approximation and comparison for motion by mean curvature with intersection points abstract consider the motion of a curve in the plane under its mean curvature it is a very interesting problem to investigate what happens when there are intersection points on the curve at which the mean curvature is singular in this paper we study this issue numerically by solving the allen cahn equation and the nonlocal evolution equation with kac potential the allen cahn equation is discretized by a monotone scheme and the nonlocal evolution equation with kac potential is discretized by the spectral method several curves with intersection points under motion by mean curvature are studied from a simple analysis and our numerical results we find that which direction to split of the curve at the intersection point depends on the angle of the curve at the point i e it splits in horizontal direction when the angle 2 in vertical direction when

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

algebraic function fields over finite fields with many rational places algebraic function fields or equivalently algebraic curves provide a useful tool for coding theory for instance algebraic geometric codes and trace codes but also for other branches of information theory in these applications the number of rational places of a function field plays a crucial role one is particularly interested in function fields having a large number of rational places after a short introduction into the mathematical theory of algebraic functions the paper gives a survey of old and new results on the number of rational places of function fields

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

the complete list of genera of quotients of the fq2 maximal hermitian curve for q 1 mod 4 abstract let f q 2 be the finite field with q 2 elements most of the known f q 2 maximal curves arise as quotient curves of the f q 2 maximal hermitian curve h q after a seminal paper by garcia stichtenoth and xing many papers have provided genera of quotients of h q but their complete determination is a challenging open problem in this paper we determine completely the spectrum of genera of quotients of h q for any q 1 mod 4

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

a new family of maximal curves over a finite field it has been known for a long time that the deligne lusztig curves associated to the algebraic groups of type 2a 2 2b 2 and 2g 2 defined over the finite field mathbb f n all have the maximum number of mathbb f n rational points allowed by the weil explicit formulas and that these curves are mathbb f q 2 maximal curves over infinitely many algebraic extensions mathbb f q 2 of mathbb f n serre showed that an mathbb f q 2 rational curve which is mathbb f q 2 covered by an mathbb f q 2 maximal curve is also mathbb f q 2 maximal this has posed the problem of the existence of mathbb f q 2 maximal curves other than the deligne lusztig curves and their mathbb f q 2 subcovers see for instance garcia on curves with many rational points over finite fields in finite fields with applications to coding theory cryptography and related areas pp 152 163 springer berlin 2002 and garcia and stichtenoth a maximal curve which is not a galois subcover of the hermitan curve bull braz math soc n s 37 139 152 2006 in this paper a positive answer to this problem is obtained for every q n 3 with n p r 2 p 2 prime we give a simple explicit construction of an mathbb f q 2 maximal curve mathcal x that is not mathbb f q 2 covered by any mathbb f q 2 maximal deligne lusztig curve furthermore the mathbb f q 2 automorphism group aut mathcal x has size n 3 n 3 1 n 2 1 n 2 n 1 interestingly mathcal x has a very large mathbb f q 2 automorphism group with respect to its genus g frac 1 2 n 3 1 n 2 2 1

on the pattern formation mechanism of 2 1 d max plus models we propose a max plus equation which reproduces evolutional patterns often observed in reaction diffusion systems of excitable media the equation gives a travelling wave a target pattern and a spiral pattern from appropriate initial data moreover using the advantages of max plus equations we obtain the solutions exactly by a reduction from the high dimensional equation to a lower one in the reduction we use coordinate curves according to a pattern shape it is interesting that all patterns satisfy the same reduced equation we also propose two other models similar to the previous one and discuss the behaviour of solutions

periodic travelling waves in a family of deterministic cellular automata abstract reaction diffusion equations whose kinetics contain a stable limit cycle are an established class of models for a range of biological and chemical systems in this paper i construct a family of deterministic cellular automata with nine states which are qualitatively similar to oscillatory reaction diffusion equations in that their rules reflect both local oscillations and spatial diffusion the automata can be crudely interpreted as models of predator prey interactions and i show that the behaviour following local perturbation of the prey only state in one space dimension is very similar in the automata and in standard reaction diffusion models for predator prey systems in particular in many cases invasion of prey by predators leaves behind periodic travelling waves in the wake of invasion i study in detail these periodic plane waves in the automaton by explicitly investigating periodic solutions of the difference equation governing travelling waves i show that the automaton has many different periodic wave solutions and i compare their properties with those of periodic wave solutions of reaction diffusion systems the basic conclusion is that included amongst the periodic waves in the automaton are a family of solutions which mimic quite closely the properties of reaction diffusion periodic waves

analysis of meandering in zykov kinetics abstract an analysis of computer results of zykov kinetics is given the meandering of the spiral tip is modeled by a composition of two circular motions one of them is the circle along which the spiral tip would move if the spirals were rotating rigidly the other one is the circle along which the spiral tip moves around a point of the previous circle this model suggests that the meandering of the spiral tip develops from the rigid rotation through a hopf bifurcation

stability of vortex rotation in an excitable cellular medium abstract stability of a vortex was studied in a modified cellular automation model the model includes dependence of wave velocity of excitation interval and front curvature two kinds of vortex instability are observed as found also in models based on partial differential equations meandering of the vortex and vortex drift in a heterogeneous medium

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

an isotropic cellular automaton for excitable media abstract we propose a new cellular automaton ca model which reproduces isotropic time evolution patterns observed in the belousov zhabotinsky reaction although several ca models have been proposed exhibiting isotropic patterns of the reaction most of them need complicated rules and a large number of neighboring cells our model can produce isotropic patterns from a simple probabilistic rule among a few 4 or 8 neighboring cells

algebraic curves with many automorphisms abstract let x be a projective geometrically irreducible non singular algebraic curve of genus g 2 defined over an algebraically closed field k of odd characteristic p let aut x be the group of all automorphisms of x which fix k element wise it is known that if aut x 8 g 3 then the p rank equivalently the hasse witt invariant of x is zero this raises the problem of determining the minimum value function f g such that whenever aut x f g then x has zero p rank for even g we prove that f g 900 g 2 the odd genus case appears to be much more difficult although for any genus g 2 if aut x has a solvable subgroup g such that g 252 g 2 then x has zero p rank and g fixes a point of x our proofs use the hurwitz genus formula and the deuring shafarevich formula together with a few deep results from group theory characterizing finite simple groups whose sylow 2 subgroups have a cyclic subgroup of index 2 we also point out some connections with the abhyankar conjecture and the katz gabber covers

convolution generated motion as a link between cellular automata and continuum pattern dynamics cellular automata have been used to model the formation and dynamics of patterns in a variety of chemical biological and ecological systems however for patterns in which sharp interfaces form and propagate automata simulations can exhibit undesirable properties including spurious anisotropy and poor representation of interface curvature effects these simulations are also prohibitively slow when high accuracy is required even in two dimensions also the highly discrete nature of automata models makes theoretical analysis difficult in this paper we present a method for generating interface motions that is similar to the threshold dynamics type cellular automata but based on continuous convolutions rather than discrete sums these convolution generated motions naturally achieve the fine grid limit of the corresponding automata and they are also well suited to numerical and theoretical analysis because of this the desired pattern dynamics can be computed accurately and efficiently using adaptive resolution and fast fourier transform techniques and for a large class of convolutions the limiting interface motion laws can be derived analytically thus convolution generated motion provides a numerically and analytically tractable link between cellular automata models and the smooth features of pattern dynamics this is useful both as a means of describing the continuum limits of automata and as an independent foundation for expressing models for pattern dynamics in this latter role it also has a number of benefits over the traditional reaction diffusion ginzburg landau continuum pde models of pattern formation which yield true moving interfaces only as singular limits we illustrate the power of this approach with convolution generated motion models for pattern dynamics in developmental biology and excitable media

branching pattern formation that reflects the history of signal propagation in living organisms branching structures are often observed in open systems during the process of structure formation deformation signal propagation can be observed branching paths often deform depending on the history of signal propagation to gain a better understanding of the process of pattern formation that results in characteristic geometrical paths we adopt a system in which the dynamics of path formation are correlated with signal propagation this model involves both branch generation dynamics and signal propagation dynamics and we introduced positive feedback between these two dynamic processes we studied the geometrical properties of path deformation and the pattern of signal propagation using a discretized reaction diffusion model the proposed model can qualitatively reproduce different branching patterns and means of signal propagation one remarkable result is that the mutual interaction of these two dynamic processes leads to autonomous wave generation similar to a pacemaker or the generation of spiral waves because the autonomous wave generation in the signal is spontaneous the shapes of the branching paths become distorted we discuss the correlation between path deformation and signal propagation as a first step in understanding signal processing for such complex deformable paths

on curves covered by the hermitian curve abstract for each proper divisordof q q 1 withqan even power of a prime maximal curves of genus 1 2 q q 1 d 1 that are f q covered by the hermitian curve are constructed

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

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 model for fast computer simulation of waves in excitable media abstract starting from a two variable system of reaction diffusion equations an algorithm is devised for efficient simulation of waves in excitable media the spatio temporal resolution of the simulation can be varied continuously for fine resolutions the algorithm provides accurate solution of the underlying reaction diffusion equations for coarse resolutions the algorithm provides qualitative simulations at small computational cost

spatiotemporal structures in a model with delay and diffusion pattern formation described by differential difference equations with diffusion is investigated it is shown that an arbitrarily small diffusion induces space time turbulence just at the instability threshold of the homogeneous stationary solution we prove this property by deriving a complex ginzburg landau equation on the basis of normal form analysis well above threshold such turbulent structures give way to synchronized states ordered by spirals and targets this secondary instability can be understood with an asymptotic method representing the system as a cellular automaton network

classification of contractively complemented hilbertian operator spaces abstract we construct some separable infinite dimensional homogeneous hilbertian operator spaces h m r and h m l which generalize the row and column spaces r and c the case m 0 we show that a separable infinite dimensional hilbertian jc triple is completely isometric to one of h m r h m l h m r h m l or the space spanned by creation operators on the full anti symmetric fock space in fact we show that h m l respectively h m r is completely isometric to the space of creation respectively annihilation operators on the m respectively m 1 anti symmetric tensors of the hilbert space together with the finite dimensional case studied in m neal b russo representation of contractively complemented hilbertian operator spaces on the fock space proc amer math soc 134 2006 475 485 this gives a full operator space classification of all rank one jc triples in terms of creation and annihilation operator spaces we use the above structural result for hilbertian jc triples to show that all contractive projections on a c algebra a with infinite dimensional hilbertian range are expansions which we define precisely of normal contractive projections from a onto a hilbertian space which is completely isometric to r c r c or this generalizes the well known result first proved for b h by robertson in a g robertson injective matricial hilbert spaces math proc cambridge philos soc 110 1991 183 190 that all hilbertian operator spaces that are completely contractively complemented in a c algebra are completely isometric to r or c we use the above representation on the fock space to compute various completely bounded banach mazur distances between these spaces or

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

contractive projections and operator spaces parallel to the study of finite dimensional banach spaces there is a growing interest in the corresponding local theory of operator spaces we define a family of hilbertian operator spaces h k n 1 k n generalizing the row and column hilbert spaces r n and c n and we show that an atomic subspace x c b h that is the range of a contractive projection on b h is isometrically completely contractive to an sum of the h k n and cartan factors of types 1 to 4 in particular for finite dimensional x this answers a question posed by oikhberg and rosenthal explicit in the proof is a classification up to complete isometry of atomic w closed jw triples without an infinite dimensional rank 1 w closed ideal

a cellular automaton describing the formation of spatially ordered structures in chemical systems abstract a cellular automaton describing a certain heterogeneous catalytic reaction is introduced as a theoretical approach towards the investigation of pattern formation in chemical systems the numerically obtained results demonstrate that the time evolution of this cellular automaton leads to a self sustained organization of circular and spiral wave like structures even starting from random initial states

diffusion and wave propagation in cellular automaton models of excitable media in this paper the authors 1 examine general masks as discrete approximations to the diffusion equation showing how to calculate the diffusion coefficient from the elements of the mask 2 combine the mask with a thresholding operation to simulate the propagation of waves in excitable media showing that for well chosen masks the waves obey a linear speed curvature relation with slope given by the predicted diffusion coefficient and 3 assess the utility of different masks in terms of computational efficiency and adherence to a linear speed curvature relation

third generation cellular automaton for modeling excitable media abstract this paper introduces a new cellular automation model of excitable media with improved treatments of 1 diffusion and wave propagation and 2 slow dynamics of the recovery variable the automaton is both computationally efficient and faithful to the underlying partial differential equations

theory of reentrant excitation in a ring of cardiac tissue a cardiac pathology reentrant tachycardia is caused by the sustained circulation of activation along a reentrant path in cardiac tissue a new theoretical model for wave propagation in discrete excitable media employing coupled maps with continuous time is used to derive analytical stability criteria for the uni directional circulation of the excitation pulse on the ring of cardiac tissue numerical simulations of the model that incorporate experimentally measurable characteristics of the cardiac tissue reproduce the experimental data quantitatively the complex oscillations in the cycle length of unstable tachycardia observed experimentally are quasiperiodic oscillations that arise via the multiple hopf bifurcation in high dimensional maps

dispersion of traveling waves in the belousov zhabotinskii reaction abstract we use perturbation and numerical methods to calculate the dispersion relation for periodic wave train solutions of diffusion reaction equations modeling the belousov zhabotinskii reaction these results are compared with experimental data for the belousov zhabotinskii reagent and shown to be in good agreement

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

patterns of conductivity in excitable automata with updatable intervals of excitations we define a cellular automaton where a resting cell excites if number of its excited neighbors belong to some specified interval and boundaries of the interval change depending on ratio of excited and refractory neighbors in the cell s neighborhood we calculate excitability of a cell as a number of possible neighborhood configurations that excite the resting cell we call cells with maximal values of excitability conductive in exhaustive search of functions of excitation interval updates we select functions which lead to formation of connected configurations of conductive cells the functions discovered are used to design conductive wirelike pathways in initially nonconductive arrays of cells we demonstrate that by positioning seeds of growing conductive pathways it is possible to implement a wide range of routing operations including reflection of wires stopping wires formation of conductive bridges and generation of new wires in the result of collision the findings presented may be applied in designing conductive circuits in excitable nonlinear media reaction diffusion chemical systems neural tissue and assemblies of conductive polymers