lagrangian subvarieties of the moduli space of stable vector bundles on a regular algebraic surface with pg 0 mukai m1 showed that there is a nondegenerate symplectic structures on the moduli space of stable vector bundles on a k3 surface later tyurin t2 studied generalized symplectic structures on the moduli space of stable vector bundles on any smooth regular surface x with p9 0 in the work of tyurin a symplectic structure means a nonzero regular two form on the moduli spaces in particular it may degenerate in this paper we define a lagrangian subvariety of the moduli space to be a subvariety on the zariski tangent space at any point of which the given symplectic two form is identically zero note that we do not impose any restriction on the dimension of a lagrangian subvariety this is because symplectic structures considered here may degenerate the purpose of this paper is to use bril l noether theory for curves to construct explicit ly a family of lagrangian subvarieties of the moduli space of stable vector bundles on a regular surface with pg 0 le t p be a generically smooth and irreducible component of the moduli space of gieseker stable bundles of rank r 1 with respect to a fixed polarization d on a regular algebraic surface x with p 0 by the boundedness of jf see ma after possibly twisted by the same negative line bundle on x we can assume that for any point e c j i e is generated by global sections ii hl e hz e hi e o iii h l de t e hz det e 0 for any point e c z choose a r 1 dimensional subspace v c h and consider an evaluation map e v v p x e for a general v we can make e v degenerate exactly along a smooth curve c c x and coker e v is a line bundle

error analysis of nonconforming and mixed fems for second order linear non selfadjoint and indefinite elliptic problems the state of the art proof of a global inf sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite second order linear elliptic pdes this paper therefore first analyses a nonconforming finite element discretization which converges owing to some a priori l 2 l2 error estimates even for reduced regularity on non convex polygonal domains an equivalence result of that nonconforming finite element scheme to the mixed finite element method mfem leads to the well posedness of the discrete solution and to a priori error estimates for the mfem the explicit residual based a posteriori error analysis allows some reliable and efficient error control and motivates some adaptive discretization which improves the empirical convergence rates in three computational benchmarks

geometry of curves with exceptional secant planes linear series along the general curve we study linear series on a general curve of genus g whose images are exceptional with regard to their secant planes working in the framework of an extension of brill noether theory to pairs of linear series we prove that a general curve has no linear series with exceptional secant planes in a very precise sense whenever the total number of series is finite next we partially solve the problem of computing the number of linear series with exceptional secant planes in a one parameter family in terms of tautological classes associated with the family by evaluating our hypothetical formula along judiciously chosen test families as an application we compute the number of linear series with exceptional secant planes on a general curve equipped with a one dimensional family of linear series we pay special attention to the extremal case of d secant d 2 planes to 2d 1 dimensional series which appears in the study of hilbert schemes of points on surfaces in that case our formula may be rewritten in terms of hypergeometric series which allows us both to prove that it is nonzero and to deduce its asymptotics in d

convergence of adaptive finite element methods for general second order linear elliptic pdes we prove convergence of adaptive finite element methods afems for general nonsymmetric second order linear elliptic pdes thereby extending the result of morin nochetto and siebert it siam j numer anal 38 2000 pp 466 488 it siam rev 44 2002 pp 631 658 the proof relies on quasi orthogonality which accounts for the bilinear form not being a scalar product together with novel error and oscillation reduction estimates which now do not decouple we show that afems are a contraction for the sum of energy error plus oscillation numerical experiments including oscillatory coefficients and both coercive and noncoercive convection diffusion pde illustrate the theory and yield optimal meshes

brill noether theory of curves on enriques surfaces i the positive cone and gonality we study the existence of linear series on curves lying on an enriques surface and general in their complete linear system using a method that works also below the bogomolov reider range we compute in all cases the gonality of such curves we also give a new result about the positive cone of line bundles on an enriques surface and we show how this relates to the gonality

three dimensional dissipative structures in reaction diffusion systems abstract we study analytically and numerically the tridimensional pattern selection problem for reaction diffusion systems qualitative agreement is found with the recent experimental results

the generic green lazarsfeld secant conjecture using lattice theory on special k3 surfaces calculations on moduli stacks of pointed curves and voisin s proof of green s conjecture on syzygies of canonical curves we prove the prym green conjecture on the naturality of the resolution of a general prym canonical curve of odd genus as well as many cases of the green lazarsfeld secant conjecture on syzygies of non special line bundles on general curves

a unifying theory of a posteriori finite element error control residual based a posteriori error estimates are derived within a unified setting for lowest order conforming nonconforming and mixed finite element schemes the various residuals are identified for all techniques and problems as the operator norm l of a linear functional of the form in the variable of a sobolev space v the main assumption is that the first order finite element space is included in the kernel ker l of l as a consequence any residual estimator that is a computable bound of l can be used within the proposed frame without further analysis for nonconforming or mixed fe schemes applications are given for the laplace stokes and navier lame equations

a unifying theory of a posteriori error control for nonconforming finite element methods residual based a posteriori error estimates were derived within one unifying framework for lowest order conforming nonconforming and mixed finite element schemes in carstensen numer math 100 617 637 2005 therein the key assumption is that the conforming first order finite element space v h c annulates the linear and bounded residual l written v h c subseteq rm ker ell that excludes particular nonconforming finite element methods ncfems on parallelograms in that v h c not subset rm ker ell the present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates also for mortar and discontinuous galerkin methods the key assumption is the existence of some bounded linear operator pi v h c rightarrow v h nc with some elementary properties it is conjectured that the more general hypothesis h1 h3 can be established for all known ncfems applications on various nonstandard finite element schemes for the laplace stokes and navier lame equations illustrate the presented unifying theory of a posteriori error control for ncfem

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

linear systems on k3 sections on etudie la presence d un g d r special sur une section d hyperplan donnee c d une surface k 3 et les consequences sur l existence d un tel g d r sur toute courbe non singuliere c e c

from quasi 2d to 3d turing patterns in ramped systems abstract we elaborate on the transition from quasi two dimensional to three dimensional turing patterns in a chemical reaction diffusion system confined in gradients of chemicals between two feed boundaries this transition is observed in open spatial reactors specially designed to make possible the unfolding of a pattern sequence in one direction of the plane of observation in this direction the confinement of the structure is progressively relaxed complementary observations from two reactor geometries allow the dimensionality of the structure to be elucidated quasi two dimensional and three dimensional patterns respectively correspond to patterns developing in monolayers and in bilayers beyond the now classical hexagonal and stripe patterns various new stable planforms are shown to result from the coupling of these two classical pattern modes which develop in two adjacent layers with well defined phase relations between the two pattern modes

convergence and domain decomposition algorithm for nonconforming and mixed methods for nonselfadjoint and indefinite problems in this paper the existence uniqueness and convergence of the solutions of nonconforming and mixed methods with non quasi uniform partition are proven for nonselfadjoint and indefinite second order elliptic problems under minimal regularity assumption moreover uniform convergence of the solution of nonconforming method is obtained meanwhile the domain decomposition algorithm which suits non quasi uniform partition is proposed for nonconforming and mixed methods

on the nef cone of symmetric products of a generic curve the nef cone of the symmetric product c of a curve c with general moduli is easily determined when k is at least equal to the gonality gon c of the curve in this paper we describe the nef cone for k gon c 1 when the genus of c is even in this case the boundary of the nef cone has rational slope and is determined using curves on c associated to the g s of c 1 introduction let c be a smooth irreducible curve of genus g 1 for any k 2 denote by c its mi symmetric product that parametrizes effective degree k divisors z x x2 on c let ni c k resp nl c k be the set of r linear combinations of 1 cycles resp divisors on k modulo numerical equivalence nx k is called the n ron severi group of c and denoted ns c k n k and nl k are finite dimensional vector spaces and the bilinear form given by the intersection pairing is nondegenerate if c has general moduli n c and nl c have rank two an interesting problem is to try to determine inside n c the convex cone of classes of effective 1 cycles denoted by ne c and called the cone of curves or mori cone dually one can study inside nl c the nef cone nef k i e the closure of the convex cone of classes of ample divisors by kleiman s criterion of ampleness

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

generalized lazarsfeld mukai bundles and a conjecture of donagi and morrison abstract let s be a k3 surface and assume for simplicity that it does not contain any 2 curve using coherent systems we express every non simple lazarsfeld mukai bundle on s as an extension of two sheaves of some special type that we refer to as generalized lazarsfeld mukai bundles this has interesting consequences concerning the brill noether theory of curves c lying on s from now on let g denote the genus of c and a be a complete linear series of type g d r on c such that d g 1 and the corresponding brill noether number is negative first we focus on the cases where a computes the clifford index if r 1 and with only some completely classified exceptions we show that a coincides with the restriction to c of a line bundle on s this is a refinement of green and lazarsfeld s result on the constancy of the clifford index of curves moving in the same linear system then we study a conjecture of donagi and morrison predicting that under no hypothesis on its clifford index a is contained in a g e s which is cut out from a line bundle on s and satisfies e g 1 we provide counterexamples to the last inequality already for r 2 a slight modification of the conjecture which holds for r 1 2 is proved under some hypotheses on the pair c a and its deformations we show that the result is optimal in the sense that our hypotheses cannot be avoided by exhibiting in appendix a some counterexamples obtained jointly with andreas leopold knutsen

turing patterns in three dimensions we investigate three dimensional turing patterns in two component reaction diffusion systems the fitzhugh nagumo equation the brusselator and the gray scott model are solved numerically in three dimensions several interconnected structures of domains as well as lamellar hexagonal and spherical domains are obtained as stable motionless equilibrium patterns the relative stability of these structures is studied analytically based on the reduction approximation the relation with the microphase separated structures in block copolymers is also discussed

the geometry of the moduli space of odd spin curves the spin moduli spacesg is the parameter space of theta characteristics spin structures on stable curves of genus g it has two connected components s g ands g depending on the parity of the spin structure we establish a complete birational classication by kodaira dimension of the odd components g of the spin moduli space we show thats g is uniruled for g 12 and even unirational for g 8 in this range introducing the concept of cluster for the mukai variety whose one dimensional linear sections are general canonical curves of genus g we construct new birational models ofs g these we then use to explicitly describe the birational structure of s g for instance s 8 is birational to a locally trivial p 7 bundle over the moduli space of elliptic curves with seven pairs of marked points for g 12 we prove thats g is a variety of general type in genus 12 this requires the construction of a counterexample to the slope conjecture on eective divisors on the moduli space of stable curves of genus 12

numerical study of three dimensional turing patterns using a meshless method based on moving kriging element free galerkin efg approach in this paper a numerical procedure is presented for solving a class of three dimensional turing system first we discrete the spatial direction using element free galerkin efg method based on the shape functions of moving kriging interpolation then to achieve a high order accuracy we use the fourth order exponential time differencing runge kutta etdrk4 method using this discretization for the temporal dimension we obtain an explicit scheme and do not need to solve nonlinear system of equations the efg method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the efg method test and trial functions are moving least squares approximation mls shape functions since the shape functions of moving least squares mls approximation do not have kronecker delta property we cannot implement the essential boundary condition directly also building shape functions of mls approximation is a time consuming procedure because of the mentioned reasons we employ the shape functions of moving kriging interpolation technique which have the mentioned property and less cpu time is required for building them for testing this method on three dimensional pdes we select some equations and system of pdes such as allen cahn gray scott ginzburg landau brusselator models predator prey model with additional food supply to predator several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme

a stabilised finite element method for the convection diffusion reaction equation in mixed form abstract this paper is devoted to the approximation of the convection diffusion reaction equation using a mixed first order formulation we propose and analyse a stabilised finite element method that allows equal order interpolations for the primal and dual variables this formulation reminiscent of the galerkin least squares method is proven stable and convergent in addition a numerical assessment of the numerical performance of different stabilised finite element methods for the mixed formulation is carried out and the different methods are compared in terms of accuracy stability and sharpness of the layers for two different classical test problems

a uniformly accurate finite element method for the reissner mindlin plate a simple finite element method for the reissner mindlin plate model in the prim itive variables is presented and analyzed the method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging it is proved that the method converges with optimal order uniformly with respect to thickness

irreducibility brill noether loci and vojta s inequality this paper deals with generalizations of hilbert s irreducibility theorem the classical hilbert irreducibility theorem states that for any cover f of the projective line defined over a number field k there exist infinitely many k rational points on the projective line such that the fiber of f over p is irreducible over k in this paper we consider similar statements about algebraic points of higher degree on curves of any genus we prove that hilbert s irreducibility theorem admits a natural generalization to rational points on an elliptic curve and thus via a theorem of abramovich and harris to points of degree 3 or less on any curve we also present examples that show that this generalization does not hold for points of degree 4 or more these examples come from an earlier geometric construction of debarre and fahlaoui some additional necessary facts about this construction can be found in the appendix provided by debarre we exhibit a connection between these irreducibility questions and the sharpness of vojta s inequality for algebraic points on curves in particular we show that vojta s inequality is not sharp for the algebraic points arising in our examples

two level additive schwarz preconditioners for nonconforming finite element methods two level additive schwarz preconditioners are developed for the nonconforming p1 finite element approximation of scalar second order symmetric positive definite elliptic boundary value problems the morley finite element approximation of the biharmonic equation and the divergence free nonconforming p1 finite element approximation of the stationary stokes equations the condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap

interconnected turing patterns in three dimensions we study numerically the turing pattern in three dimensions in a fitzhugh nagumo type reaction diffusion system we have found that interconnected periodic domain structures such as a gyroid fddd and perforated lamellar structures appear in three dimensions which never exist in lower dimensions the stability analysis of these structures is also performed by means of a mode expansion

higher dimensional localized patterns in excitable media abstract the excitable reaction diffusion equation model of the form e u t e 2 2 u u v v t 2 v u v is considered when u is assumed to be of mckean s piecewise linear type the interfacial approach can be applied to the stability of various localized patterns in higher dimensional spaces it is shown that a band shaped localized pattern is destabilized into a zig zag mode or a varicose mode and that a disk shaped localized pattern is destabilized into static modes and when is small into an oscillatory mode like a breather motion numerical simulations are performed to confirm such destabilizations for more general nonlinear functions u

pattern fluid interpretation of chemical turbulence the spontaneous formation of heterogeneous patterns is a hallmark of many nonlinear systems from biological tissue to evolutionary population dynamics the standard model for pattern formation in general and for turing patterns in chemical reaction diffusion systems in particular are deterministic nonlinear partial differential equations where an unstable homogeneous solution gives way to a stable heterogeneous pattern however these models fail to fully explain the experimental observation of turbulent patterns with spatio temporal disorder in chemical systems here we introduce a pattern fluid model as a general concept where turbulence is interpreted as a weakly interacting ensemble obtained by random superposition of stationary solutions to the underlying reaction diffusion system the transition from turbulent to stationary patterns is then interpreted as a condensation phenomenon where the nonlinearity forces one single mode to dominate the ensemble this model leads to better reproduction of the experimental concentration profiles for the stationary phases and reproduces the turbulent chemical patterns observed by q ouyang and h l swinney chaos 1 411 1991

some new error estimates for ritz galerkin methods with minimal regularity assumptions new uniform error estimates are established for finite element approximations u h of solutions u of second order elliptic equations lu f using only the regularity assumption u 1 c f 1 using an aubin nitsche type duality argument we show for example that for arbitrary fixed e sufficiently small there exists an h 0 such that for 0 h h 0 u u h 0 e u u h 1 here s denotes the norm on the sobolev space h s other related results are established

convergence and quasi optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems recently an adaptive nonconforming finite element method anfem has been developed by carstensen and hoppe in numer math 103 251 266 2006 in this paper we extend the result to some nonsymmetric and indefinite problems the main tools in our analysis are a posteriori error estimators and a quasi orthogonality property in this case we need to overcome two main difficulties one stems from the nonconformity of the finite element space the other is how to handle the effect of a nonsymmetric and indefinite bilinear form an appropriate adaptive nonconforming finite element method featuring a marking strategy based on the comparison of the a posteriori error estimator and a volume term is proposed for the lowest order crouzeix raviart element it is shown that the anfem is a contraction for the sum of the energy error and a scaled volume term between two consecutive adaptive loops moreover quasi optimality in the sense of quasi optimal algorithmic complexity can be shown for the anfem the results of numerical experiments confirm the theoretical findings

spinodal decomposition in 3 space using a cell dynamical system cds to model the separation of phases we present the results of large scale simulations of spinodal decomposition in 3 space of a symmetric binary alloy and incompressible binary fluid at critical quench our main results are 1 the reliable determination of the asymptotic or the almost asymptotic form factors for these systems and 2 an understanding of the preasymptotic behavior of growth laws in terms of dispersion relations of interface fluctuations to achieve 1 it is indispensable to have i methods to analyze the data ii a scheme to simulate a system with fluid dynamic interactions and iii a demonstration of the self averaging nature of spherically averaged quantities item iii allows us to study sufficiently large systems with currently available computational resources a careful stability analysis of our cds model such as artificial pinning anisotropy etc is also given we must point out that the conventional data analysis coupled with artificial pinning and finite size effects may spuriously give the theoretically desired results especially in binary fluids the asymptotic form factors are estimated in collage form from our large scale late time simulation results and various theoretical asymptotic results such as tomita s sum rule we may conclude that for binary fluids the agreement between our form factor and ones obtained experimentally is excellent this also clearly demonstrates that apart from time and space scale changes binary polymer systems so far studied and low molecular weight systems are not distinct that is no polymer effect has been observed in the form factor in the case of binary alloys the agreement with our form factor and various experimental results is not as good as the binary fluid case this may be ascribed to various complications in solids anisotropy elastic effects etc

moduli of theta characteristics via nikulin surfaces we study moduli spaces of k3 surfaces endowed with a nikulin involution and their image in the moduli space r g of prym curves of genus g we observe a striking analogy with mukai s well known work on ordinary k3 surfaces many of mukai s results have a very precise prym nikulin analogue for instance a general prym curve from r g is a section of a nikulin surface if and only if g 7 and g 6 furthermore r 7 has the structure of a fibre space over the corresponding moduli space of polarized nikulin surfaces we then use these results to study the geometry of the moduli space s g of even spin curves with special emphasis on the transition case of s 8 which is a 21 dimensional calabi yau variety

the mercat conjecture for stable rank 2 vector bundles on generic curves we prove the mercat conjecture for rank 2 vector bundles on generic curves of every genus for odd genus we identify the effective divisor in mg where the mercat conjecture fails and compute its slope the clifford index of a smooth curve c defined as the non negative quantity cliff c min deg l 2r l l pic c h c l 2 deg l g 1 stratifies the moduli space mg of curves of genus g with the smallest stratum being that consisting of hyperelliptic curves which have clifford index zero while the definition is obviously inspired by the classical clifford theorem the notion came to real prominence in the works of green lazarsfeld g gl voisin v2 and many others in the context of syzygies of canonical curves for a general curve c of genus g one has cliff c bg 1 2 c the locus of curves c mg with cliff c bg 1 2 c is a subvariety of codimension 1 respectively 2 when g is odd respectively even much studied by harris and mumford hm it has been a long standing problem to find an adequate definition of the clifford index for higher rank vector bundles on curves a higher rank clifford index should not only capture the behavior of the generic curve from the point of view of special higher rank vector bundles but also provide a geometrically meaningful stratification ofmg an interesting definition has been put forward by lange and newstead ln1 for a semistable vector bundle e of rank 2 and slope e on a curve c of genus g one defines its clifford index as cliff e e h c e 2 0 the rank 2 clifford index of c is then defined as the quantity cliff2 c min cliff e e uc 2 d d 2g 2 h c e 4 observe that if l is a line bundle on c with deg l 2h0 c l 2 cliff c that is l computes the clifford index of c then cliff l l cliff c in particular the inequality cliff2 c cliff c holds for every curve c of genus g the main achievement of this paper is the proof of the following result theorem 1 for a general curve c of genus g the following equality holds cliff2 c cliff c g 1 2 equivalently if e is a semistable rank 2 vector bundle on c contributing to cliff2 c then h c e e 2 cliff c mercat me conjectured the equality cliff2 c cliff c for every smooth curve c mg counterexamples to this expectation have been found using noether lefschetz special k3 surfaces in fo1 fo2 and ln2 however these examples are special in moduli and theorem 1 proves mercat s conjecture for generic curves of every genus

projectivity and birational geometry of bridgeland moduli spaces we construct a family of nef divisor classes on every moduli space of stable complexes in the sense of bridgeland this divisor class varies naturally with the bridgeland stability condition for a generic stability condition on a k3 surface we prove that this class is ample thereby generalizing a result of minamide yanagida and yoshioka our result also gives a systematic explanation of the relation between wall crossing for bridgeland stability and the minimal model program for the moduli space we give three applications of our method for classical moduli spaces of sheaves on a k3 surface 1 we obtain a region in the ample cone in the moduli space of gieseker stable sheaves only depending on the lattice of the k3 2 we determine the nef cone of the hilbert scheme of n points on a k3 surface of picard rank one when n is large compared to the genus 3 we verify the hassett tschinkel huybrechts sawon conjecture on the existence of a birational lagrangian fibration for the hilbert scheme in a new family of cases

singularities of moduli spaces of sheaves on k3 surfaces and nakajima quiver varieties abstract the aim of this paper is to study the singularities of certain moduli spaces of sheaves on k3 surfaces by means of nakajima quiver varieties the singularities in question arise from the choice of a non generic polarization with respect to which we consider stability and admit natural symplectic resolutions corresponding to choices of general polarizations for sheaves that are pure of dimension one we show that these moduli spaces are locally around a singular point isomorphic to a quiver variety and that via this isomorphism the natural symplectic resolutions correspond to variations of git quotients of the quiver variety

pattern formation in the three dimensional reaction diffusion systems existing group theoretic analysis of pattern formation in three dimensions t k callahan e knobloch symmetry breaking bifurcations on cubic lattices nonlinearity 10 1997 1179 1216 is used to make specific predictions about the formation of three dimensional patterns in two models of the turing instability the brusselator model and the lengyel epstein model spatially periodic patterns having the periodicity of the simple cubic sc face centered cubic fcc or body centered cubic bcc lattices are considered an efficient center manifold reduction is described and used to identify parameter regimes permitting stable lamellae sc fcc double diamond hexagonal prism bcc and bcci states both models possess a special wavenumber k at which the normal form coefficients take on fixed model independent ratios and both are described by identical bifurcation diagrams this property is generic for two species chemical reaction diffusion models with a single activator and inhibitor 1999 elsevier science b v all rights reserved

morphological transitions and bistability in turing systems it is well known that in two dimensions turing systems produce spots stripes and labyrinthine patterns and in three dimensions lamellar and spherical structures or their combinations are observed in this paper we study transitions between these states in both two and three dimensions first we derive the regions of stability for different patterns using nonlinear bifurcation analysis then we apply large scale computer simulations to analyze the pattern selection in a bistable system by studying the effect of parameter selection on morphological clustering and the appearance of topological defects the method elaborated in this paper presents a probabilistic approach for studying pattern selection in a bistable reaction diffusion system

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

a new dimension to turing patterns abstract it is well known that simple reaction diffusion systems can display very rich pattern formation behaviour here we have studied two examples of such systems in three dimensions first we investigate the morphology and stability of a generic turing system in three dimensions and then the well known gray scott model in the latter case we added a small number of morphogen sources in the system in order to study its robustness and the formation of connections between the sources our results raise the question of whether turing patterning can produce an inductive signalling mechanism for neuronal growth