extended mean field study of a stochastic cellular automaton extended mean field theory is used for studying the one dimensional stochastic cellular automaton with rule 18 defined by wolfram rev mod phys 55 601 1983 the analysis is carried out at different levels taking n point and n pair correlations explicitly into consideration the pair approximations reproduce the exact results in the deterministic limit the critical behavior is studied by the pad e approximant method and the predicted critical probability and exponent agree with previous data within a few percent

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

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

weak majorization inequalities and convex functions abstract let f be a convex function defined on an interval i 0 1 and a b n n complex hermitian matrices with spectrum in i we prove that the eigenvalues of f a 1 b are weakly majorized by the eigenvalues of f a 1 f b further if f is log convex we prove that the eigenvalues of f a 1 b are weakly majorized by the eigenvalues of f a f b 1 as applications we obtain generalizations of the famous golden thomson trace inequality a representation theorem and a harmonic geometric mean inequality some related inequalities are discussed

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

some inequalities of majorization type abstract we show some majorization inequalities and apply them to derive norm eigenvalue singular value and trace inequalities of matrices we also present a generalization and a different proof of a recent result on majorization concerning partitioned hermitian positive semidefinite matrices

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

constrained control and approximation properties of a rational interpolating curve this paper deals with the convexity control and the strain energy control of interpolating curves using a rational cubic spline with linear denominator the sufficient and necessary conditions for controlling the interpolating curve to be convex or concave are derived when the function being interpolated is f t c 3 t 0 t n the error estimation of the interpolating function and the boundedness of the optimal error coefficient and its double symmetry with regard to parameters are obtained

oscillation and pattern formation in a system of self regulating cells abstract in this paper a model of a system of n 2 asymmetrically and nondiffusibly coupled self regulating cells is proposed intended to formally represent well known basic features of biological systems organization the model is discussed in terms of the activation desensitization dynamics in dictyostelium it is shown that the cells in the ensemble are capable of stable oscillations provided the intercellular coupling is strong enough otherwise their stationary values remain asymptotically stable the emergence of oscillations in the one dimensional ring shaped ensemble of size n 2 is due to a bifurcation of naimark sacker type which is proven to exist under rather mild conditions it is shown that a supercritical as well as a subcritical bifurcation branch exist at different parameter values moreover due to the dihedral symmetry of the system the parity of n is shown to determine the multiplicity of the critical pair of eigenvalues crossing the unit circle the model proposed does not represent a reaction diffusion system it is not excitable but exhibits pattern formation under conditions under which a stable limit cycle exists

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 bivariate rational interpolation and the properties abstract in this paper a bivariate rational interpolation is constructed using both function values and partial derivatives of the function being interpolated as the interpolation data the interpolation function has a simple and explicit rational mathematical representation with parameters and it can be expressed by the symmetric bases it is proved that the interpolation is stable the concept of integral weights coefficients of the interpolation is given which describes the weight of the interpolation points and the quantity as the interpolation data in the local interpolating region

more results on singular value inequalities of matrices abstract the arithmetic geometric mean inequality for singular values due to bhatia and kittaneh says that 2 s j ab s j a a b b j 1 2 for any matrices a b we give a new equivalent form and some relevant generalizations of this inequality in particular we show that s j a 1 4 b 3 4 a 3 4 b 1 4 s j a b j 1 n for any n n positive semidefinite matrices a b which proves a special case of zhan s conjecture posed in 2000

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

weighted rational cubic spline interpolation and its application abstract in qi duan et al korean j comput appl math 6 1 1999 203 215 the authors have discussed constrained interpolation problems by means of rational cubic spline interpolation with linear denominators but there are still some cases in which the constrained interpolation cannot be solved in this paper the weighted rational cubic spline interpolation has been constructed using the rational cubic spline with linear denominator and the rational cubic spline based on function values by these the problems to constrain the weighted rational interpolation curves to lie strictly above or below a given piecewise linear curve and between two given piecewise linear curves can be solved completely also the approximation properties of these weighted rational cubic splines are studied

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

on the evolution of periodic plane waves in reaction diffusion systems of l w type lambda omega systems are a class of simple reaction diffusion equations with a limit cycle in the reaction kinetics the author considers the solution of the system given by lambda r lambda 0 r p omega r omega 0 r p on a semi infinite spatial domain with initial data decaying exponentially across the domain numerical evidence is presented showing that this initial condition induces a wave front moving across the domain with periodic plane waves behind the front these periodic waves can move in either direction depending on the parameter values the author uses intuitive criteria to derive an expression for the speed of the advancing front and by reducing the system to ordinary differential equations for similarity solutions the amplitude and speed of the periodic plane waves are determined the speed of these periodic waves varies continuously with the initial data perturbation theory is then used to obtain an analytical approximation to the solutions these anal

singular values of convex functions of operators and the arithmetic geometric mean inequality abstract we prove singular value inequalities for convex functions of products and sums of operators that generalize the arithmetic geometric mean inequality for operators among other results we prove that if a i b i x i y i i 1 n are operators on a complex separable hilbert space such that x i 2 y i 2 2 n i i 1 n and if f is a nonnegative increasing convex function on 0 satisfying f 0 0 then s j f i 1 n a i x i y i b i 1 2 s j k 1 n x k i 1 n f a i a k x k y k i 1 n f b i b k y k for j 1 2

inequalities for sums and products of operators abstract let ai i 1 4 be compact operators on a complex separable hilbert space we show that 2 s j a 1 a 2 a 3 a 4 1 2 s j a 1 a 2 a 3 a 4 for j 1 2 from this we derive norm inequalities and investigate conditions for equality

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

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

cellular automation model of reaction transport processes abstract the transport and chemical reactions of solutes are modelled as a cellular automation in which molecules of different species perform a random walk on a regular lattice and react according to a local probabilistic rule the model describes advection and diffusion in a simple way and as no restriction is placed on the number of particles at a lattice site it is also able to describe a wide variety of chemical reactions assuming molecular chaos and a smooth density function we obtain the standard reaction transport equations in the continuum limit simulations on one an two dimensional lattices show that the discrete model can be used to approximate the solutions of the continuum equations we discuss discrepancies which arise from correlations between molecules and how these disappear as the continuum limit is approached of particular interest are simulations displaying long time behaviour which depends on long wavelength statistical fluctuations not accounted for by the standard equations the model is applied to the reactions a b c and a b c with homogenous and inhomogenous initial conditions as well as to systems subject to autocatalytic reactions and displaying spontaneous formation of spatial concentration patterns

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

a bivariate rational cubic interpolating spline with biquadratic denominator a bivariate rational bicubic interpolating spline bris with biquadratic denominator and six shape parameters is constructed in a rectangle domain the c1 continuous condition of bris discussed bris is proved to be bounded and its error is estimated in the case of the equally spaced knots the matrix expression and symmetry of bris are presented some properties of the basis of bris are given in the end a numerical example is given to illustrat the effect of the shape parameters on the shape of bris surface

hermite interpolation by piecewise rational surface in this paper a bivariate rational hermite interpolation is developed to create a space surface using both function values and the first order partial derivatives of the function being interpolated as the interpolation data the interpolation function has a simple and explicit rational mathematical representation with parameters there are two schemes to construct the interpolation functions an example shows that both of them approximate the function being interpolated very well the basis of this interpolation is derived and the interpolating function can be c1 in all the interpolation region when the parameters satisfy a simple condition when a patch of the interpolating surface does not satisfy the need of design say it is too high or too low at a point and its neighbourhood the values of the interpolating function in these points need to be decreased or increased under the condition that the interpolating data are not changed for this sufficient conditions are derived and it is shown that this can be done just by selecting suitable parameters in the interpolation

critical behaviour of a probabilistic automata network sis model for the spread of an infectious disease in a population of moving individuals a probabilistic automata network sis susceptible infective susceptible model for the spread of an infectious disease in a population of moving individuals is studied the local rule consists of two subrules the first one applied synchronously models infection and recovery it is a probabilistic cellular automaton rule the second applied sequentially describes the motion of the individuals the model contains three parameters the probabilities pi to get infected and pr to recover and the average number of tentative moves per individual m depending upon the values of these parameters in the infinite time limit the system is either in the disease free state or in the endemic state it goes from one state to the other through a transcritical bifurcation similar to a second order phase transition characterized by a non negative order parameter whose role is played in this model by the stationary density of infected individuals the pi pr phase diagram and the critical behaviour of the stationary density of infectives in the neighbourhood of the phase transition are studied as a function of m according to whether the individuals perform short or long range moves it is found that the parameters characterizing the transition have a qualitatively different behaviour as m varies when m is very large the correlations created by the application of the subrule modelling infection and recovery are destroyed and as expected the behaviour of the system is then correctly predicted by a mean field type approximation which assumes a homogeneous mixing of the individuals when m is not large this assumption is no longer correct

irregular wakes in reaction diffusion waves abstract reaction diffusion equations have proved to be highly successful models for a wide range of biological and chemical systems but chaotic solutions have been very rarely documented we present a new mechanism for generating apparently chaotic spatiotemporal irregularity in such systems by analysing in detail the bifurcation structure of a particular set of reaction diffusion equations on an infinite one dimensional domain with particular initial conditions we show that possible solutions include travelling fronts which leave behind either regular or irregular spatiotemporal oscillations using a combination of analytical and numerical analysis we show that the irregular behaviour arises from the instability of oscillations induced by the passage of the front finally we discuss the generality of this mechanism as a way in which spatiotemporal irregularities can arise naturally in reaction diffusion systems

wavelengths distribution of chaotic travelling waves in some cellular automata abstract travelling waves tw solutions under the dynamics of one dimensional infinite cellular automata ca exist abundantly in many cases we show that for any permutative ca unstable tw are dense in the space of configurations then we consider the cases where the number of states is a prime number so that the state space is a finite field k and the automata rules are linear on k we give an algorithm for the computation of the tw for any integer velocity of propagation larger than the interaction range then we show that their wavelengths are characterized in terms of zeros of an associated family of polynomials over k and we describe the mathematical complexity of wavelengths distributions in various linear ca laws we also obtain some exponential lower bound for the growth of the number of waves in terms of the velocity in rule 90

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

more inequalities for positive semidefinite 2 2 block matrices and their blocks abstract it is obtained that for positive semidefinite block matrix h m k k n if a b m n c such that m a x a 2 b 2 2 and f is a nonnegative increasing convex function on 0 satisfying f 0 0 then 2 s j f a k b m a x a 2 b 2 s j f h for j 1 2 n also it is shown that if h m k k n is a positive semidefinite 2 2 block matrix with square matrices m and n of the same size then h r 2 a b r 1 m n for r 1 where a and b are the largest eigenvalues of the matrices m and n respectively in addition if r 3 2 then h r 2 4 m 2 n 2 m 0 n 0 2 r 2 for every unitarily invariant norm

bivariate polynomial and continued fraction interpolation over ortho triples abstract by means of the barycentric coordinates expression of the interpolating polynomial over each ortho triple some properties are obtained moreover the explicit coefficients in terms of b net for one ortho triple and two ortho triples are worked out respectively thus the computation of multiple integrals can be converted into the sum of the coefficients in terms of the b net over triangular domain much effectively and conveniently based on a new symmetrical algorithm of partial inverse differences a novel continued fractions interpolation scheme is presented over arbitrary ortho triples in r2 which is a bivariate osculatory interpolation formula with one order partial derivatives at all corner points in the ortho triples furthermore its characterization theorem is presented by three term recurrence relations the new scheme is advantageous over the polynomial one with some numerical examples

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

stability of periodic travelling wave solutions with large spatial periods in reaction diffusion systems abstract we consider a spatially homogeneous system of reaction diffusion equation defined on the interval of the one dimensional spatial variable x it is known that this equation has a one parameter family of periodic travelling wave solutions x ct c if this equation has a spatially homogeneous periodic solution t the spatial period l c of the travelling wave solution satisfies l c c t if c where c is the propagation speed and t is the period of t we prove that in the case c 0 is sufficiently large x ct c is unstable if t is strongly unstable and x ct c is marginally stable if t is strongly stable if the equation is defined on a finite interval 0 l of the variable x with the periodic boundary conditions we can obtain a more precise result regarding the stability of x c t c where c 0 is a speed which satisfies l ml c for some positive integer m we prove that this solution is asymptotically stable in the sense of waveform stability if c 0 is sufficiently large and if t is strongly stable

inequalities between f a b and f a f b abstract the conjecture posed by aujla and silva j s aujla f c silva weak majorization inequalities and convex functions linear algebra appl 369 2003 217 233 is proved it is shown that for any m tuple of positive semidefinite n n complex matrices a j and for any non negative convex function f on 0 with f 0 0 the inequality f a 1 f a 2 f a m f a 1 a 2 a m holds for any unitarily invariant norm it is also proved that f a 1 f a 2 f a m f a 1 a 2 a m where f is a non negative concave function on 0 and is normalized

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

spatial organization and evolution period of the epidemic model using cellular automata we investigate epidemic models with spatial structure based on the cellular automata method the construction of the cellular automata is from the study by weimar and boon about the reaction diffusion equations phys rev e 49 1749 1994 our results show that the spatial epidemic models exhibit the spontaneous formation of irregular spiral waves at large scales within the domain of chaos moreover the irregular spiral waves grow stably the system also shows a spatial period 2 structure at one dimension outside the domain of chaos it is interesting that the spatial period 2 structure will break and transform into a spatial synchronous configuration in the domain of chaos our results confirm that populations embed and disperse more stably in space than they do in nonspatial counterparts

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

eigenvalue majorization inequalities for positive semidefinite block matrices and their blocks abstract let h m k k n be a positive semidefinite block matrix with square matrices m and n of the same order and denote i 1 the main results are the following eigenvalue majorization inequalities for any complex number z of modulus 1 h 1 2 m n i z k z k o 1 2 m n i z k z k o if in addition k is hermitian then for any real number r 2 2 h 1 2 m n r k o 1 2 m n r k o while if k is skew hermitian then for any real number r 2 2 h 1 2 m n r i k o 1 2 m n r i k o where o is the zero matrix of compatible size these majorization inequalities generalize some results due to furuichi and lin turkmen paksoy and zhang lin and wolkowicz

two types of performance of an isotropic cellular automaton stationary turing patterns and spiral waves abstract an isotropic cellular automaton containing only the crudest caricature of activator inhibitor systems is shown to be capable of producing stationary spatial structures turing patterns in qualitative agreement with a variety of experiments in chemistry due to the isotropy of our automaton we can obtain spots arranged in regular hexagons a pattern that cannot be obtained with a classical square lattice labyrinthine stripes like those observed in magnetic domains in thin films are also obtained spiral waves are obtained upon increasing the radius of action of the activator relative to that of the inhibitor as predicted by theory

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

notes on matrix arithmetic geometric mean inequalities abstract for positive semi definite n n matrices the inequality 4 ab a b 2 isshown to hold for every unitarily invariant norm the connection of this with some other matrix arithmetic geometric mean inequalities and trace inequalities is discussed