self similar solutions to a density dependent reaction diffusion model in this paper we investigated a density dependent reaction diffusion equation u t u m xx u u m this equation is known as the extension of the fisher or kolmogoroff petrovsky piscounoff equation which is widely used in population dynamics combustion theory and plasma physics by employing a suitable transformation this equation was mapped to the anomalous diffusion equation where the nonlinear reaction term was eliminated due to its simpler form some exact self similar solutions with compact support have been obtained the solutions evolving from an initial state converge to the usual traveling wave at a certain transition time hence the connection between the self similar solution and the traveling wave solution is quite clear from these results moreover the solutions were found in a manner that propagates either to the right or to the left furthermore the two solutions form a symmetric solution expanding in both directions applications to spatiotemporal pattern formation in biological populations is discussed

dual processes in neural network models i neural dynamics versus dynamics of learning the dynamics of learning in an unsupervised formulation of the kohonen model is shown to be equivalent to the dynamics of local order parameters in an attractor neural network with short range hebbian interactions and long range anti hebbian interactions this duality can be used to study analytically the creation of topology conserving maps

regularizing effect of absorption terms in singular problems abstract we prove existence of solutions to problems whose model is p u u q f u in u 0 in u 0 on where is an open bounded subset of r n n 2 p u is the p laplacian operator for 1 p n q 0 0 and f is a nonnegative function in l m for some m 1 in particular we analyze the regularizing effect produced by the absorption term in order to infer the existence of finite energy solutions in case 1 we also study uniqueness of these solutions as well as examples which show the optimality of the results finally we find local w 1 p solutions in case 1

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

reflection group of the quaternionic lorentzian leech lattice we study a second example of the phenomenon studied in the article the complex lorentzian leech lattice and the bimonster we find 14 roots in the automorphism group of the quaternionic lorentzian leech lattice l that form the coxeter diagram given by the incidence graph of projective plane over f2 we prove that the reflections in these roots generate the automorphism group of l the investigation is guided by an analogy with the theory of weyl groups there is a unique point in the quaternionic hyperbolic space fixed by the diagram automorphisms that we call the weyl vector the unit multiples of the 14 roots forming the diagram are the analogs of the simple roots the 14 mirrors perpendicular to the simple roots are the mirrors that are closest to the weyl vector

phase transitions in simple learning the authors investigate learning in the simplest type of a layered neural network the one layer perceptron the learning process is treated as a statistical dynamical problem quantities one is interested in include the relaxation time the learning time and the capacity and how they depend on noise and constraints on the weights the relaxation time is calculated as a function of the noise level and the number p of associations to be learned they consider three different cases for input patterns that are random and uncorrelated in the first where the connection weights are constrained to satisfy n 1 sigma i omega i2 s2 there is a critical value of p n separating regimes of perfect and imperfect learning at zero noise in contrast the second model unconstrained learning exhibits a different kind of transition at p n and noise plays no role in the third model where the constraint is imposed only on the thermal fluctuations there is a line of phase transitions terminating at p n and zero noise they have also considered learning with correlated input patterns the most important difference is the emergence of a second relaxation time which the authors interpret as the time it takes to learn a prototype of the patterns

the complex lorentzian leech lattice and the bimonster ii let d be the incidence graph of the projective plane over f3 the artin group of the graph d maps onto the bimonster and a complex hyperbolic reflection group acting on 13 dimensional complex hyperbolic space y the generators of the artin group are mapped to elements of order 2 resp 3 in the bimonster resp let y y be the complement of the union of the fixed points of reflections in daniel allcock has conjectured that the orbifold fundamental group of y surjects onto bimonster in this article we study the reflection group we show that th e artin group of d maps to the orbifold fundamental group of y thus answering a question in allcock s article a monstrous proposal and taking one step towards the proof of allcock s conjecture the finite group aut d acts on y we make a detailed study of the complex hyperbolic line fixed by the subgroup l3 3 aut d we define meromorphic automorphic forms on y invariant under with poles along mirrors we show that the restriction of these forms to the complex hyperbolic line fixed by l3 3 gives meromorphic modular forms of level 13

the connections of large perceptrons the authors derive analytical expressions for the connections of large perceptrons by studying the fixed points of the perceptron learning rule if the training set consists of all possible input vectors they can calculate for large systems the connections as a series expansion in the system size the leading term in this expansion turns out to be either the hebb rule for unbiased distributions or the biased hebb rule for biased distributions the performance of their asymptotic expressions and finite size corrections on small systems is studied numerically for the more realistic case of having an extensive training set patterns learned with training noise they derive a self consistent set of coupled nonlinear equations for the connections in the limit of zero training noise the solution of these equations is shown to give the connections with maximal stability in the gardner sense

a convergent numerical scheme for integrodifferential kinetic models of angiogenesis abstract we study a robust finite difference scheme for integrodifferential kinetic systems of fokker planck type modeling tumor driven blood vessel growth the scheme is of order one and enjoys positivity features we analyze stability and convergence properties and show that soliton like asymptotic solutions are correctly captured we also find good agreement with the solution of the original stochastic model from which the deterministic kinetic equations are derived working with ensemble averages a numerical study clarifies the influence of velocity cut offs on the solutions for exponentially decaying data

the effect of turbulence on mixing in prototype reaction diffusion systems the effect of turbulence on mixing in prototype reaction diffusion systems is analyzed here in the special situation where the turbulence is modeled ideally with two separated scales consisting of a large scale mean flow plus a small scale spatiotemporal periodic flow in the limit of fast reaction and slow diffusion it is rigorously proved that the turbulence does not contribute to the location of the mixing zone in the limit and that this mixing zone location is determined solely by advection of the large scale velocity field this surprising result contrasts strongly with earlier work of the authors that always yields a large scale propagation speed enhanced by small scale turbulence for propagating fronts the mathematical reasons for these differences are pointed out this main theorem rigorously justifies the limit equilibrium approximations utilized in nonpremixed turbulent diffusion flames and condensation evaporation modeling in cloud physics in the fast reaction limit the subtle nature of this result is emphasized by explicit examples presented in the fast reaction and zero diffusion limit with a nontrivial effect of turbulence on mixing in the limit the situation with slow reaction and slow diffusion is also studied in the present work here the strong stirring by turbulence before significant reaction occurs necessarily leads to a homogenized limit with the strong mixing effects of turbulence expressed by a rigorous turbulent diffusivity modifying the reaction diffusion equations physical examples from non premixed turbulent combustion and cloud microphysics modeling are utilized throughout the paper to motivate and interpret the mathematical results c 2000 john wiley sons inc

on the y555 complex reflection group abstract we give a computer free proof of a theorem of basak describing the group generated by 16 complex reflections of order 3 satisfying the braid and commutation relations of the y 555 diagram the group is the full isometry group of a certain lattice of signature 13 1 over the eisenstein integers z 1 3 along the way we enumerate the cusps of this lattice and classify the root and niemeier lattices over z 1 3

anzellotti s pairing theory and the gauss green theorem abstract in this paper we obtain a very general gauss green formula for weakly differentiable functions and sets of finite perimeter this result is obtained by revisiting anzellotti s pairing theory and by characterizing the measure pairing a d u when a is a bounded divergence measure vector field and u is a bounded function of bounded variation

complex reflection groups braid groups hecke algebras presentations a la coxeter are given for all irreducible nite com plex re ection groups they provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra hecke algebras we introduce monodromy representations of the braid groups which factorize through the hecke algebras extending results of cherednik opdam kohno and others summary introduction complex re ection groups and their presentations a background from complex re ection groups b presentations braid groups and their diagrams a generalities about hyperplane complements b generalities about the braid groups c the braid diagrams proofs of the main theorems for the braid groups b de e r a notation and prerequisites b computation of b de e r and of its center for d c computation of b e e r and of its center hecke algebras a background from di erential equations and monodromy b a family of monodromy representations of the braid group c hecke algebras diagrams and tables appendix generators of the monodromy around a divisor appendix tables to mathematics subject classi cation primary g we thank jean michel peter orlik pierre vogel for useful conversations and the isaac newton institute for its hospitality while the last version of this manuscript was written up the second named author gratefully acknowledges nancial support by the fondation alexander von humboldt for his stays in paris michel brou e gunter malle raphael rouquier

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

learning and forgetting on asymmetric diluted neural networks it is possible to construct diluted asymmetric models of neural networks for which the dynamics can be calculated exactly we test several learning schemes in particular models for which the values of the synapses remain bounded and depend on the history our analytical results on the relative efficiencies of the various learning schemes are qualitatively similar to the corresponding ones obtained numerically on fully connected symmetric networks

locally toroidal polytopes of rank 6 and sporadic groups abstract we augment the list of finite universal locally toroidal regular polytopes of type 3 3 4 3 3 due to p mcmullen and e schulte adding as well as removing entries this disproves a related long standing conjecture our new universal polytope is related to a well known y shaped presentation for the sporadic simple group fi 22 and admits s 4 o 8 2 s 3 as the automorphism group we also discuss further extensions of its quotients in the context of y shaped presentations as well we note that two known examples of finite universal polytopes of type 3 3 4 3 3 are related to y shaped presentations of orthogonal groups over f 2 mixing construction is used in a number of places to describe covers and 2 covers

a singular reaction diffusion system modelling prey predator interactions invasion and co extinction waves abstract we consider a singular reaction diffusion system arising in modelling prey predator interactions in a fragile environment since the underlying odes system exhibits a complex dynamics including possible finite time quenching one first provides a suitable notion of global travelling wave weak solution then our study focusses on the existence of travelling waves solutions for predator invasion in such environments we devise a regularized problem to prove the existence of travelling wave solutions for predator invasion followed by a possible co extinction tail for both species under suitable assumptions on the diffusion coefficients and on species growth rates we show that travelling wave solutions are actually positive on a half line and identically zero elsewhere such a property arising for every admissible wave speeds

sharp profiles in degenerate and doubly degenerate fisher kpp equations this paper investigates the effects of a degenerate diffusion term in reaction diffusion models ut d u ux x g u with fisher kpp type g both in the case when d 0 0 and when d 0 d 1 0 with d u 0 elsewhere we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c and we show the appearance of a sharp type profile when c c these results solve recent conjectures formulated by sanchez garduno and maini j differential equations 117 1995 281 and satnoianu et al discrete continuous dyn systems series b 1 2000 339

unsupervised dynamic learning in layered neural networks the authors consider a stochastic two layer neural network of binary neurons in which the connections between the layers are updated according to the hebb rule whereas the lateral connections in the output layer are modified according to an anti hebb rule in equilibrium the output overlap is found to be a linear transformation of the input overlap next they extend the model by considering learning as a dynamic process which means that synaptic efficacies as well as neuronal states may vary in time despite the coupling of these two variables they show that in this particular model the behaviour can be well analysed it turns out that the network filters the information available at the input in such a way that important components of the input data can pass through whereas components with a low information content are suppressed

qualitative behavior and traveling waves for flux saturated porous media equations arising in optimal mass transportation abstract this paper is devoted to the analysis of qualitative properties of porous media flux saturated type operators which combine the effects of reaction terms of fkpp type in dimension one specifically we study the existence of traveling wave profiles sub and super solutions waiting time and dynamical properties of the support the paper also deals with the connection between these minimal traveling waves with those of other related models porous media flux saturated mechanism and the heat equation

stability of travelling waves for non convex scalar viscous conservation laws travelling waves of a viscous conservation law so called viscous profiles are shown to be stable in polynomialy weighted l spaces there is no assumption of convexity on the nonlinear term and thus earlier results of il in and ole nik sattinger and kawashima and matsumura are generalized the method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates are derived for the semigroup through a new technique for estimating the resolvent 1993 john wiley sons inc

short and long range connections in autoassociative memory the authors consider memory retrieval in a network of m modules a module consists of n neuronal units each of which is connected to all n 1 other units within the same module and to l units distributed randomly throughout all the other modules both short and long range connections are symmetric the units are threshold linear with a continuous positive output each module can retrieve one of d local activity patterns or features stored on the corresponding short range connections furthermore p global activity patterns each consisting of combinations of m local features are stored on the dilute long range connections when m 1 the long range connections endow the network with attractor states correlated with a single global pattern and they study its storage capacity within a mean field approach if p d and each feature appears in only one pattern their model reduces to an intermediate case between fully connected and highly dilute architectures whose capacities they recover in the appropriate limits as p d takes larger integer values the maximum p grows but it remains asymptotically proportional to n rather than to l n 1 the total number of connections per unit the maximum amount of retrievable information per synapse on the other hand decreases moreover as p d grows retrieval attractors have to compete with a memory glass state involving the retrieval of spurious combinations of features whose existence and stability they describe analytically they suggest implications for neocortical memory functions

on the relativistic heat equation in one space dimension we study the relativistic heat equation in one space dimension we prove a local regularity result when the initial datum is locally lipschitz in its support we propose a numerical scheme that captures the known features of the solutions and allows for analysing further properties of their qualitative behaviour

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a general fractional porous medium equation we develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion u t 2 u m 1 u 0 x r n t 0 u x 0 f x x r n we consider data f l 1 r n and all exponents 0 0 existence and uniqueness of a weak solution is established for m m n n giving rise to an l 1 contraction semigroup in addition we obtain the main qualitative properties of these solutions in the lower range 0 m m existence and uniqueness of solutions with good properties happen under some restrictions and the properties are different from the case above m we also study the dependence of solutions on f m and moreover we consider the above questions for the problem posed in a bounded domain

fisher equation with density dependent diffusion special solutions a special family of solutions explicit in the space and time variables is found for the fisher equation with density dependent diffusion the connection with the known travelling wave solution and the initial conditions from which that evolves is also shown

the 1 harmonic flow with values into a smooth planar curve abstract we introduce a notion of solution to the 1 harmonic flow i e the formal gradient flow of the total variation with respect to the l 2 distance from a domain of r m into a connected subset of the image of a smooth jordan curve for such notion we establish existence and uniqueness of solutions to the homogeneous neumann problem we also discuss a consistent notion of solution when the target space is a smooth n 1 dimensional manifold whose geodesics are unique presenting conjectures and open questions related to it

convergence to travelling waves for quasilinear fisher kpp type equations we consider the cauchy problem ut u xx u t x r r u 0 x u0 x x r when the increasing function satisfies that 0 0 and the equation may degenerate at u 0 in the case of 0 0 we consider the case of u0 l r 0 u0 x 1 a e x r and the special case of u u u we prove that the solution approaches the travelling wave solution with speed c 1 spreading either to the right or to the left or to the two travelling waves moving in opposite directions

an extension of the pairing theory between divergence measure fields and bv functions abstract in this paper we introduce a nonlinear version of the notion of anzellotti s pairing between divergence measure vector fields and functions of bounded variation motivated by possible applications to evolutionary quasilinear problems as a consequence of our analysis we prove a generalized gauss green formula

travelling waves and finite propagation in a reaction diffusion equation abstract we study the existence of travelling wave solutions and the property of finite propagation for the reaction diffusion equation ut um xx un x t r 0 with m 1 0 n r and u u x t 0 we show that travelling waves exist globally only if m n 2 and only for velocities c c 2 m in the study of propagation we must take into account that solutions of the cauchy problem can be nonunique for n c 0 for m n 2 maximal solutions propagate instantly to the whole space if n

pattern formation in a flux limited reaction diffusion equation of porous media type a non linear pde featuring flux limitation effects together with those of the porous media equation non linear fokker planck is presented in this paper we analyze the balance of such diverse effects through the study of the existence and qualitative behavior of some admissible patterns namely traveling wave solutions to this singular reaction diffusion equation we show the existence and qualitative behavior of different types of traveling waves classical profiles for wave speeds high enough and discontinuous waves that are reminiscent of hyperbolic shock waves when the wave speed lowers below a certain threshold some of these solutions are of particular relevance as they provide models by which the whole solution and not just the bulk of it as it is the case with classical traveling waves spreads through the medium with finite speed

a class of extended limiters applied to piecewise hyperbolic methods piecewise hyperbolic methods phms have been widely used for the approximation of hyperbolic conservation laws in this paper we have studied the power limiters in order to analyze their suitable use in the design of phms we formulate a new phm based on these limiters improving the behavior of the phm in siam j sci comput 15 1994 pp 892 915 at local extrema jump discontinuities and discontinuities in derivative we show that this new hyperbolic reconstruction we call it powerphm is local total variation bounded and it behaves stably with a reduced numerical viscosity compared to phm we show numerical evidence of the above mentioned features by means of one and two dimensional numerical tests with the euler equations of gas dynamics

on the entropy conditions for some flux limited diffusion equations abstract in this paper we give a characterization of the notion of entropy solutions of some flux limited diffusion equations for which we can prove that the solution is a function of bounded variation in space and time this includes the case of the so called relativistic heat equation and some generalizations for them we prove that the jump set consists of fronts that propagate at the speed given by rankine hugoniot condition and we give on it a geometric characterization of the entropy conditions since entropy solutions are functions of bounded variation in space once the initial condition is to complete the program we study the time regularity of solutions of the relativistic heat equation under some conditions on the initial datum an analogous result holds for some other related equations without additional assumptions on the initial condition

the speed of propagation for kpp type problems ii general domains this paper is devoted to nonlinear propagation phenomena in general unbounded domains of rn for reaction diffusion equations with kolmogorov petrovsky piskunov kpp type nonlinearities this article is the second in a series of two and it is the follow up of the paper 7 which dealt which the case of periodic domains this paper is concerned with general domains and we give various definitions of the spreading speeds at large times for solutions with compactly supported initial data we study the relationships between these new notions and analyze their dependence on the geometry of the domain and on the initial condition some a priori bounds are proved for large classes of domains the case of exterior domains is also discussed in detail lastly some domains which are very thin at infinity and for which the spreading speeds are infinite are exhibited the construction is based on some new heat kernel estimates in such domains

effect of noise on front propagation in reaction diffusion equations of kpp type we consider reaction diffusion equations of kpp type in one spatial dimension perturbed by a fisher wright white noise under the assumption of uniqueness in distribution examples include the randomly perturbed fisher kpp equations partial tu partial x 2u u 1 u epsilon sqrt u 1 u dot w and partial tu partial x 2u u 1 u epsilon sqrt u dot w where dot w dot w t x is a space time white noise we prove the brunet derrida conjecture that the speed of traveling fronts for small e is 2 pi 2 log epsilon 2 2 o log log epsilon log epsilon 3

the automorphism group of the 26 dimensional even unimodular lorentzian lattice abstract the group mentioned in the title is shown to be a certain infinitely generated coxeter group extended by the negative of the identity operation and the group of all automorphisms of the notorious leech lattice

on the fkpp equation with gaussian shear advection abstract a boundary value problem for the fisher kolmogorov petrovskii piskunov fkpp equation with shear random advection is investigated it is demonstrated that the upper bounds for front propagation in the fkpp equation with gaussian advection obtained by an analysis of its linear ensemble averaged upper solution can give too rough predictions it is shown that the unboundedness of the gaussian advection affects ensemble averaged solutions of linear and nonlinear problems in a different way this analytical prediction is confirmed by direct numerical simulations the phenomenon of propagation failure due to zero boundary conditions is studied and the critical conditions are found numerical procedure for the analysis of random wave propagation is suggested by introduction of mean front position and its variance some numerical experiments are presented

the fisher kpp problem with doubly nonlinear diffusion abstract the famous fisher kpp reaction diffusion model combines linear diffusion with the typical kpp reaction term and appears in a number of relevant applications in biology and chemistry it is remarkable as a mathematical model since it possesses a family of travelling waves that describe the asymptotic behaviour of a large class solutions 0 u x t 1 of the problem posed in the real line the existence of propagation waves with finite speed has been confirmed in some related models and disproved in others we investigate here the corresponding theory when the linear diffusion is replaced by the slow doubly nonlinear diffusion and we find travelling waves that represent the wave propagation of more general solutions even when we extend the study to several space dimensions a similar study is performed in the critical case that we call pseudo linear i e when the operator is still nonlinear but has homogeneity one with respect to the classical model and the pseudo linear case the slow travelling waves exhibit free boundaries

a variational principle for kpp front speeds in temporally random shear flows we establish the variational principle of kolmogorov petrovsky piskunov kpp front speeds in temporally random shear flows with sufficiently decaying correlations a key quantity in the variational principle is the almost sure lyapunov exponent of a heat operator with random potential to prove the variational principle we use the comparison principle of solutions the path integral representation of solutions and large deviation estimates of the associated stochastic flows the variational principle then allows us to analytically bound the front speeds the speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length and the sublinear growth law if the temporal decorrelation is also large enough the so called bending phenomenon

y groups via transitive extension abstract motivated by an earlier observation by j h fischer around 1980 b conway conjectured that the coxeter diagram y555 on fig 1 together with a single additional so called spider relation ab1c1ab2c2ab3c3 10 1 form a presentation for the wreath product of the monster group m and a group of order 2 this conjecture was proved by s p norton and the author in 1990 the original proof was rather involved relying on simple connectedness results for certain diagram geometries on numerous data obtained by coset enumeration on a computer and on some delicate calculations with subgroups coming from the 26 node theorem in the present work we follow an inductive approach to the identification of y groups by considering larger y groups as transitive extensions of smaller ones along these lines we obtain an alternative identification proof for the y groups which is almost computer free we refer to only one result of double coset enumeration our approach provides a uniform understanding of the y groups particularly of features such as centres and redundant generators

competition between pattern reconstruction and sequence processing in nonsymmetric neural networks the authors study an ising spin neural network model in which the interaction matrix consists of a symmetric hebbian term which favours the reconstruction of static patterns and a nonsymmetric transition term which favours limit cycles corresponding to the processing of pattern sequences they calculate phase diagrams and analyse the relation between the relative weight of the two competing contributions to the interaction matrix and the frequency of the periodic attractors

global travelling waves in reaction convection diffusion equations we study the existence and properties of solutions in travelling wave form u x t x st defined for every z x st r for the reaction convection diffusion equationut a um xx b un x kupforx r t 0 with a m n 0 b k p r in the reaction case k 0 we prove that there exist travelling waves vanishing for z if and only if b 0 and min 1 n m p2 max 1 n p max 1 n moreover if m p 2n there exists a minimal velocity s a b k m n p 0 for which there are travelling waves only with s s while in the case m p 2n there are travelling waves only when 4amk b2n and for every velocity s 0 some properties of the function s are established all the waves are decreasing in their support and waves having bounded support from the right exist if and only if m min 1 n also the absorption case k 0 is treated where we find that for different values of the parameters there exists a unique travelling wave for every velocity s r but for some case where only negative velocities exist the cases b 0 or k 0 are well known in the literature

on burgers type equations with nonmonotonic dissipative fluxes we study a formation of patterns in burgers type equations endowed with a bounded but nonmonotonic dissipative flux ut f u x q ux x q s s 1 s 2 issues of uniqueness existence and smoothness of a solution are addressed asymptotic regions of a solution are discussed in particular classical and nonclassical traveling waves with an embedded subshock are constructed c 1998 john wiley sons inc

reductions and symmetries for a generalized fisher equation with a diffusion term dependent on density and space abstract in this work a generalized fisher equation with a space density diffusion term is analyzed by applying the theory of symmetry reductions for partial differential equations the study of this equation is relevant in terms of its applicability in cell dynamics and tumor invasion therefore classical lie symmetries admitted by the equation are determined in addition by using the multipliers method we derive some nontrivial conservation laws for this equation finally we obtain a direct reduction of order of the ordinary differential equations associated and a particular solution

26 implies the bimonster the bimonster or wreathed square 2 of the monster group is presented by the coxeter relations of the 26 node incidence graph of the projective plane of order 3 along with the additional relations that all free 12 gons of this diagram generate symmetric groups s we prove this 12 using purely geometric arguments as well as the celebrated ivanov norton theorem 6 7 9 this elementary presentation provides a natural connection from the projective plane of order 3 to the bimonster previously this connection was known in the other direction 2 3 we also consider other groups that can be obtained in a similar fashion a useful reference on coxeter and reflection groups is 5 we use the atlas 1 notation for finite groups

local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws this paper constructs a local third order accurate shock capturing method for hyperbolic scalar conservation laws based on numerical fluxes with a total variation diminishing tvd runge kutta evolution in time using the idea recently introduced by c w shu and s j osher for essentially nonoscillatory eno methods the constructed method is an upwind conservative scheme that is local in the sense that numerical fluxes are reconstructed without using extrapolation from the data of the smoothest neighboring cell to design the method a new concept of local smoothing is introduced to prevent the increasing of total variation of the solution near discontinuities and to achieve third order accuracy the method becomes third order accurate in smooth regions of the solution except at local extrema where it may degenerate to o h 3 2 thus giving better accuracy than tvd methods at local extrema the main advantage of this method lies on the property that is more local than that of eno and tvd upwind

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

models of collective cell spreading with variable cell aspect ratio a motivation for degenerate diffusion models continuum diffusion models are often used to represent the collective motion of cell populations most previous studies have simply used linear diffusion to represent collective cell spreading while others found that degenerate nonlinear diffusion provides a better match to experimental cell density profiles in the cell modeling literature there is no guidance available with regard to which approach is more appropriate for representing the spreading of cell populations furthermore there is no knowledge of particular experimental measurements that can be made to distinguish between situations where these two models are appropriate here we provide a link between individual based and continuum models using a multi scale approach in which we analyze the collective motion of a population of interacting agents in a generalized lattice based exclusion process for round agents that occupy a single lattice site we find that the relevant continuum description of the system is a linear diffusion equation whereas for elongated rod shaped agents that occupy l adjacent lattice sites we find that the relevant continuum description is connected to the porous media equation pme the exponent in the nonlinear diffusivity function is related to the aspect ratio of the agents our work provides a physical connection between modeling collective cell spreading and the use of either the linear diffusion equation or the pme to represent cell density profiles results suggest that when using continuum models to represent cell population spreading we should take care to account for variations in the cell aspect ratio because different aspect ratios lead to different continuum models

finite propagation speed for limited flux diffusion equations we prove that the support of solutions of a limited flux diffusion equation known as a relativistic heat equation evolves at a constant speed identified as the speed of light c for that we construct entropy sub and super solutions which are fronts evolving at speed c and prove the corresponding comparison principle between entropy solutions and sub and super solutions respectively this enables us to prove the existence of discontinuity fronts moving at light s speed

hyperbolic phenomena in a strongly degenerate parabolic equation we consider the equation u t u u x x where 0 and where is a strictly increasing function with lim s we solve the associated cauchy problem for an increasing initial function and discuss to what extent the solution behaves qualitatively like solutions of the first order conservation law u t u x equations of this type arise for example in the theory of phase transitions where the corresponding free energy functional has a linear growth rate with respect to the gradient

diffusion front capturing schemes for a class of fokker planck equations application to the relativistic heat equation in this research work we introduce and analyze an explicit conservative finite difference scheme to approximate the solution of initial boundary value problems for a class of limited diffusion fokker planck equations under homogeneous neumann boundary conditions we show stability and positivity preserving property under a courant friedrichs lewy parabolic time step restriction we focus on the relativistic heat equation as a model problem of the mentioned limited diffusion fokker planck equations we analyze its dynamics and observe the presence of a singular flux and an implicit combination of nonlinear effects that include anisotropic diffusion and hyperbolic transport we present numerical approximations of the solution of the relativistic heat equation for a set of examples in one and two dimensions including continuous initial data that develops jump discontinuities in finite time we perform the numerical experiments through a class of explicit high order accurate conservative and stable numerical schemes and a semi implicit nonlinear crank nicolson type scheme

traveling wave phenomena in some degenerate reaction diffusion equations abstract in this paper we study the existence of travelling wave solutions t w s u x t x ct for the equation formula g u where the reactive part g u is as in the fisher kpp equation and different assumptions are made on the non linear diffusion term d u both functions d and g are defined on the interval 0 1 the existence problem is analysed in the following two cases case 1 d 0 0 d u 0 u 0 1 d and g c 2 0 1 d 0 0 and d 0 0 we prove that if there exists a value of c c for which the equation possesses a travelling wave solution of sharp type it must be unique by using some continuity arguments we show that for 0 c c there are no t w s while for c c the equation has a continuum of t w s of front type the proof of uniqueness uses a monotonicity property of the solutions of a system of ordinary differential equations which is also proved case 2 d 0 d 0 0 d and g c 2 0 1 d 0 0 if in addition we impose d 0 0 with d u 0 u 0 1 we give sufficient conditions on c for the existence of t w s of front type meanwhile if d 0 d u u 0 1 we analyse just one example d u u 2 and g u u 1 u which has oscillatory t w s for 0 c 2 and t w s of front type for c 2 in both the above cases we use higher order terms in the taylor series and the centre manifold theorem in order to get the local behaviour around a non hyperbolic point of codimension one in the phase plane

effects of a saturating dissipation in burgers type equations we propose and study a new variant of the burgers equation with dissipation fluxes that saturate as the gradients become unbounded if the upstream downstream transition is above a critical threshold the corresponding riemann problem admits a weak solution wherein part of the transit is accomplished by a jump it is shown that the solution to a cauchy problem with sufficiently small compact or periodic initial data preserves its initial smoothness c 1997 john wiley sons inc

on coxeter diagrams of complex reflection groups we study complex coxeter diagrams of some unitary reflection groups using solely the combinatorics of diagrams we give a new proof of the classification of root lattices defined over e z e 2 i 3 there are only four such lattices namely the e lattices whose real forms are a2 d4 e6 and e8 next we address the issue of characterizing the diagrams for unitary reflection groups a question that was raised by broue malle and rouquier to this end we describe an algorithm which given a unitary reflection group g picks out a set of complex reflections if g is the reflection group of a root lattice defined over e the reflections selected by the algorithm form the known diagram for g for the reflection group of the complex coxeter todd lattice k e 12 we find a new diagram that extends to an affine diagram withz 7z symmetry if g is a weyl group the algorithm immediately yields a set of simple roots otherwise experimental evidences indicate that the algorithm selects a minimal generating set of reflections if g is primitive and g has a set of roots whose z span is a discrete subset of the ambient vector space

dual processes in neural network models ii analysis of zero temperature fixed point equations for pt i see ibid vol 25 p 2577 1992 the dynamics of learning in an unsupervised formulation of the kohonen model has been shown to be equivalent to the dynamics of local order parameters in an attractor neural network with short range hebbian interactions and long range anti hebbian interactions the authors analyse the zero temperature fixed point equations of these systems for the special case where the distribution of p dimensional input vectors has rotational symmetry the problem of finding the ground state is equivalent to finding the ground state of a system of p dimensional heisenberg spins with short range ferromagnetic couplings and long range anti ferromagnetic couplings

some regularity results on the relativistic heat equation we prove some partial regularity results for the entropy solution u of the so called relativistic heat equation in particular under some assumptions on the initial condition u0 we prove that ut t is a radon measure in rn moreover if u0 is log concave inside its support being a convex set then we show the solution u t is also log concave in its support t this implies its smoothness in t in that case we can give a simpler characterization of the notion of entropy solution

the leech lattice and complex hyperbolic reflections we construct a natural sequence of finite covolume reflection groups acting on the complex hyperbolic spaces h 13 h 9 and h 5 and show that the 9 dimensional example coincides with the largest of the groups of mostow 11 our reflection groups arise as automorphism groups of certain lorentzian lattices over the eisenstein integers and we obtain our largest example by using the complex leech lattice in a manner inspired by conway 5 we also construct finite covolume reflection groups on the quaternionic hyperbolic spaces h 7 h 5 and h 3 again using the leech lattice and apply results of borcherds 4 to obtain automorphic forms for our groups

a fisher kolmogorov equation with finite speed of propagation abstract in this paper we study a fisher kolmogorov type equation with a flux limited diffusion term and we prove the existence and uniqueness of finite speed moving fronts and the existence of some explicit solutions in a particular regime of the equation

uniqueness of renormalized solutions of degenerate elliptic parabolic problems we consider a general class of degenerate elliptic paraboic problems associated with the equation b v t div a v dv f using kruzhkov s method of doubling variables both in space and time we prove uniqueness and a comparison principle in l 1 for renormalized solutions

the dirichlet problem associated to the relativistic heat equation we prove existence and uniqueness of entropy solutions for the nonhomogeneous dirichlet problem associated to the relativistic heat equation

strong solutions to a parabolic equation with linear growth with respect to the gradient variable abstract in this paper we prove existence and uniqueness of strong solutions to the homogeneous neumann problem associated to a parabolic equation with linear growth with respect to the gradient variable this equation is a generalization of the time dependent minimal surface equation existence and regularity in time of the solution is proved by means of a suitable pseudoparabolic relaxed approximation of the equation and a passage to the limit

bistable traveling waves around an obstacle we consider traveling waves for a nonlinear diffusion equation with a bistable or multistable nonlinearity the goal is to study how a planar traveling front interacts with a compact obstacle that is placed in the middle of the space n as a first step we prove the existence and uniqueness of an entire solution that behaves like a planar wave front approaching from infinity and eventually reaching the obstacle this causes disturbance on the shape of the front but we show that the solution will gradually recover its planar wave profile and continue to propagate in the same direction leaving the obstacle behind whether the recovery is uniform in space is shown to depend on the shape of the obstacle 2008 wiley periodicals inc

local in time regularity results for some flux limited diffusion equations of porous media type abstract we prove some regularity results for solutions of some flux limited diffusion equations of porous media type for lipschitz initial data or assuming a uniform gradient bound on some power of the data including the fast diffusion case in which the results are global in time we also develop the existence and uniqueness theory for solutions of the fast diffusion case which was not covered in the current literature

anisotropic tempered diffusion equations abstract we introduce a functional framework which is specially suited to formulate several classes of anisotropic evolution equations of tempered diffusion type under an amenable set of hypothesis involving a very natural potential function these models can be shown to belong to the entropy solution framework devised by andreu et al 2005 therefore ensuring well posedness we connect the properties of this potential with those of the associated cost function thus providing a link with optimal transport theory and a supply of new examples of relativistic cost functions moreover we characterize the anisotropic spreading properties of these models and we determine the rankine hugoniot conditions that rule the temporal evolution of jump hypersurfaces under the given anisotropic flows

viscosity solutions with shocks a solution of single nonlinear first order equations may develop jump discontinuities even if initial data is smooth typical examples include a crude model equation describing some bunching phenomena observed in epitaxial growth of crystals as well as conservation laws where jump discontinuities are called shocks conventional theory of viscosity solutions does not apply we introduce a notion of proper viscosity solutions to track whole evolutions for such equations in multi dimensional spaces we establish several versions of comparison principles we also study the vanishing viscosity method to construct a unique global proper solution at least when the evolution is monotone in time or the initial data is monotone in some sense under additional technical assumptions in fact we prove that the graph of approximate solutions converges to that of a proper solution in the hausdorff distance topology such a convergence is also established for conservation laws with monotone data in particular local uniform convergence outside shocks is proved 2001 john wiley sons inc

a strongly degenerate quasilinear elliptic equation abstract we prove existence and uniqueness of entropy solutions for the quasilinear elliptic equation u div a u du v where 0 v l 1 r n l r n a z f z and f is a convex function of with linear growth as satisfying other additional assumptions in particular this class of equations includes the elliptic problems associated to a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics respectively in a second part of this work using crandall liggett s iteration scheme this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic cauchy problem

1 dimensional attractor neural networks we solve a class of attractor neural network models with a mixture of 1d nearest neighbour interactions and infinite range interactions which are both of a hebbian type form our solution is based on a combination of mean field methods transfer matrices and 1d random field techniques and is obtained both for boltzmann type equilibrium following sequential glauber dynamics and peretto type equilibrium following parallel dynamics competition between the alignment forces mediated via short range interactions and those mediated via infinite range ones is found to generate novel phenomena such as multiple locally stable pure states first order transitions between recall states 2 cycles and non recall states and domain formation leading to extremely long relaxation times we test our results against numerical simulations and simple benchmark cases and find excellent agreement

on the function and development of spatial structure in layered neural networks the authors study the relation between spatial structure and information processing properties of layered ising spin neural networks with lateral interactions the interactions between layers are given by the hebb rule the interactions within layers by the so called anti hebb rule secondly they study the development of spatial structure in such networks as the result of an unsupervized learning process now both neurons and interactions are dynamical variables by calculating the spectrum of the output covariance matrix as a function of the spectrum of the input covariance matrix they show how the spatial characteristics of the input signals are reflected in the information processing properties of the equilibrated system

a numerical study of turbulent flame speeds of curvature and strain g equations in cellular flows abstract we study front speeds of curvature and strain g equations arising in turbulent combustion these g equations are hamilton jacobi type level set partial differential equations pdes with non coercive hamiltonians and degenerate nonlinear second order diffusion the hamiltonian of a strain g equation is also non convex numerical computation is performed based on monotone discretization and weighted essentially nonoscillatory weno approximation of transformed g equations on a fixed periodic domain the advection field in the computation is a two dimensional hamiltonian flow consisting of a periodic array of counter rotating vortices or cellular flows depending on whether the evolution is predominantly in the hyperbolic or parabolic regimes suitable explicit and semi implicit time stepping methods are chosen the turbulent flame speeds are computed as the linear growth rates of large time solutions a new nonlinear parabolic pde is proposed for the reinitialization of level set functions to prevent piling up of multiple bundles of level sets on the periodic domain we found that the turbulent flame speed s t of the curvature g equation is enhanced as the intensity a of cellular flows increases at a rate between those of the inviscid and viscous g equations the s t of the strain g equation increases in small a decreases in larger a then drops down to zero at a large enough but finite value a the flame front ceases to propagate at this critical intensity a and is quenched by the cellular flow