descent equalities and the inductive mckay condition for types b and e abstract we establish the inductive mckay condition introduced by isaacs malle navarro 14 for finite simple groups of lie types b l l 2 e 6 e 6 2 and e 7 thus leaving open only the types d and d 2 we bring to the methods previously used by the authors for type c 8 some descent arguments using shintani s norm map this provides for types different from a d d 2 a uniform proof of the so called global requirement of the criterion given by the second author in 32 2 12 the local requirements from that criterion are verified through a detailed study of the normalizers of relevant levi subgroups and their characters

recovering asymptotics of metrics from fixed energy scattering data from fixed energy scattering data is studied it is shown that if two such metrics g1 g2 have scattering data at some fixed energy which are equal up to smoothing then there exists a diffeomorphism fixing infinity such that g1 g2 is rapidly decreasing given the scattering matrix at two energies it is shown that the asymptotics of a metric and a short range potential can be determined simultaneously these results also hold for a wide class of scattering manifolds

the computation of water waves modelled by nekrasov s equation nekrasov s integral equation describing water waves of almost extreme form is solved numerically the method consists of applying a simple quadrature rule to a rearranged version of the original equation strongly graded meshes are used to resolve an expected boundary layer in the solution for methods based on the trapezoidal rule global bifurcation theory is used to prove for fixed discretization parameter n the existence a continuous branch of positive numerical solutions these are parametrized by mu a natural parameter occurring in the original integral equation for fixed mu collective compactness arguments then prove subsequential convergence of these solutions as the mesh is refined i e as n to infty numerical experiments using higher order quadrature rules are reported these reveal that the method is capable of detecting a boundary layer and gibbs phenomenon type oscillations of maximum height about 0 37 circ in a region of width o 70 mathord left vphantom

resolvent at low energy iii the spectral measure the second and third authors were supported by australian research council discovery grants dp0771826 and dp1095448 and the second author by a future fellowship the first author was partially supported by anr grant anr 09 jcjc 0099 01 and by the pics cnrs progress in geometric analysis and applications and thanks the math department of anu for its hospitality

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

morawetz estimates for the wave equation at low frequency we consider morawetz estimates for weighted energy decay of solutions to the wave equation on scattering manifolds i e those with large conic ends we show that a morawetz estimate persists for solutions that are localized at low frequencies independent of the geometry of the compact part of the manifold we further prove a new type of morawetz estimate in this context with both hypotheses and conclusion localized inside the forward light cone this result allows us to gain a 1 2 power of t decay relative to what would be dictated by energy estimates in a small part of spacetime

notes on the wave equation on asymptotically euclidean manifolds abstract the primary object of this paper is to discuss the asymptotics of solutions of the wave equation on an asymptotically euclidean manifold when the initial data have compact support by going over to an appropriate conformal metric it is shown that just as for the ordinary wave equation such a solution has a forward future and a backward past radiation field the same method is then used to define end points of bicharacteristics that begin and end above the boundary the analogue of the sphere at infinity in the euclidean case and to derive a relation between the wave front sets of the two radiation fields the extension of these results to solutions with finite energy is also briefly discussed

operateurs conjugues et proprietes de propagation we use an algebraic criteria based on local positivity of a commutator which asserts the existence of a direction of propagation for the flowe iht associated to a self adjoint operatorh

characters of odd degree we prove the mckay conjecture on characters of odd degree a major step in the proof is the verication of the inductive mckay condition for groups of lie type and primes such that a sylow subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of harish chandra induction specics of characters of odd degree namely that most of them lie in the principal harish chandra series then allow us to deduce from it the mckay conjecture for the prime 2 hence for characters of odd degree

unipotent classes and special weyl group representations abstract we show that various invariants of a unipotent conjugacy class in a connected semisimple group can be recovered purely in terms of data involving the weyl group

action of automorphisms on irreducible characters of symplectic groups abstract assume g is a finite symplectic group sp 2 n q over a finite field f q of odd characteristic we describe the action of the automorphism group aut g on the set irr g of ordinary irreducible characters of g this description relies on the equivariance of deligne lusztig induction with respect to automorphisms we state a version of this equivariance which gives a precise way to compute the automorphism on the corresponding levi subgroup this may be of independent interest as an application we prove that the global condition in spath s criterion for the inductive mckay condition holds for the irreducible characters of sp 2 n q

pattern formation of a spatial predator prey system abstract there are random and directed movements of predator and prey populations in many natural systems which are strongly influenced and modified by spatial factors to investigate how these migration directed movement and diffusion random movement affect predator prey systems we have studied the spatiotemporal complexity in a predator prey system with holling tanner form a theoretical analysis of emerging spatial pattern is presented and wavelength and pattern speed are calculated at the same time we present the properties of pattern solutions the results of numerical simulations show that migration has prominent effect on the pattern formation of the population i e changing turing pattern into traveling pattern this study suggests that modelling by migration and diffusion in predator prey systems can account for the dynamical complexity of ecosystems

a reduction theorem for the mckay conjecture the mckay conjecture asserts that for every finite group g and every prime p the number of irreducible characters of g having p degree is equal to the number of such characters of the normalizer of a sylow p subgroup of g although this has been confirmed for large numbers of groups including for example all solvable groups and all symmetric groups no general proof has yet been found in this paper we reduce the mckay conjecture to a question about simple groups we give a list of conditions that we hope all simple groups will satisfy and we show that the mckay conjecture will hold for a finite group g if every simple group involved in g satisfies these conditions also we establish that our conditions are satisfied for the simple groups psl2 q for all prime powers q 4 and for the suzuki groups sz q and ree groups r q where q 2 e or q 3 e respectively and e 1 is odd since our conditions are also satisfied by the sporadic simple group j 1 it follows that the mckay conjecture holds for all primes p for every finite group having an abelian sylow 2 subgroup

traveling waves for a model of gravity driven film flows in cylindrical domains abstract traveling wave solutions are studied for a recently derived model of a falling viscous film on the interior of a vertical rigid tube by identifying a hopf bifurcation and using numerical continuation software families of non trivial traveling wave solutions may be traced out in parameter space these families all contain a single solution at a turnaround point with larger film thickness than all others in the family in an earlier paper it was conjectured that this turnaround point may represent a critical thickness separating two distinct flow regimes observed in physical experiments as well as two distinct types of behavior in transient solutions to the model here these hypotheses are verified over a range of parameter values using a combination of numerical and analytical techniques the linear stability of these solutions is also discussed both large and small amplitude solutions are shown to be unstable though the instability mechanisms are different for each wave type specifically for small amplitude waves the region of relatively flat film away from the localized wave crest is subject to the same instability that makes the trivial flat film solution unstable for large amplitude waves this mechanism is present but dwarfed by a much stronger tendency to relax to a regime close to that followed by small amplitude waves

the isomorphism and isogeny theorems for reductive algebraic groups let g be a connected reductive algebraic group defined over an algebraically closed field k and let t be a maximal torus of g letx be the character group oftand r subset x the root system of grelative tot thusx is isomorphic to mathbb z r with r dim t the rank of g and for each alpha in r there exists a unipotent subgroup u alpha which is isomorphic to the additive group of k and is normalized by t according to the character alpha 1 1 if u alpha k to u alpha is an isomorphism then t u alpha left a right t 1 u alpha left alpha left t right a right for all t in t and a in k

instabilities of one dimensional trivial phase solutions of the two dimensional cubic nonlinear schrodinger equation abstract the two dimensional cubic nonlinear schrodinger equation nls is used as a model of a wide variety of physical phenomena in this paper we study the stability of a class of its one dimensional periodic traveling wave solutions first we establish that all such solutions are unstable with respect to two dimensional perturbations with long wavelengths in the transverse dimension second we establish that all such solutions are unstable with respect to two dimensional perturbations with arbitrarily short wavelengths if the coefficients of the linear dispersion terms in the nls have opposite signs both arguments rely on formal perturbation methods third we use the fourier floquet hill numerical method to examine the spectral stability problem we present detailed spectra for twelve different solutions and demonstrate strong agreement between the numerical and asymptotic results

a numerical study of the large period limit of a zakharov shabat eigenvalue problem with periodic potentials deconinck and kutz 2006 j comput phys 219 296 321 developed an efficient algorithm for solving the zakharov shabat eigenvalue problem with periodic potentials numerically it is natural to use the same algorithm for solving the problem for non periodic potential decaying potentials defined over the whole real line using large periods in this paper we propose the use of a specific value of the floquet exponent our numerical results indicate that it can produce accurate results long before the period becomes large enough for the analytical convergence results of gardner 1997 j reine angew math 491 149 81 to be valid we also illustrate the rather complicated path to convergence of some nonlinear schr dinger potentials

numerical inverse scattering for the focusing and defocusing nonlinear schrodinger equations we solve the focusing and defocusing nonlinear schrodinger nls equations numerically by implementing the inverse scattering transform the computation of the scattering data and of the nls solution are both spectrally con vergent initial conditions in a suitable space are treated using the approach of biondini and bui 3 we numerically solve homogeneous robin boundary value problems on the half line finally using recent theoretical developments in the numerical approximation of riemann hilbert problems we prove that under mild assumptions our method of approximating solutions to the nls equations is uniformly accurate in their domain of definition

positive commutators at the bottom of the spectrum abstract bony and hafner have recently obtained positive commutator estimates on the laplacian in the low energy limit on asymptotically euclidean spaces these estimates can be used to prove local energy decay estimates if the metric is non trapping we simplify the proof of the estimates of bony hafner and generalize them to the setting of scattering manifolds i e manifolds with large conic ends by applying a sharp poincare inequality our main result is the positive commutator estimate i h 2 g i 2 h 2 g a i h 2 g c i h 2 g 2 where h is a large parameter i is a compact interval in 0 and i its indicator function and where a is a differential operator supported outside a compact set and equal to 1 2 r d r r d r near infinity the laplacian can also be modified by the addition of a positive potential of sufficiently rapid decay the same estimate then holds for the resulting schrodinger operator

global parametrices and dispersive estimates for variable coefficient wave equations in this article we consider variable coefficient time dependent wave equations in mathbb r times mathbb r n using phase space methods we construct outgoing parametrices and prove strichartz type estimates globally in time this is done in the context of c 2 metrics which satisfy a weak asymptotic flatness condition at infinity

convergence of hill s method for nonselfadjoint operators by the introduction of a generalized evans function defined by an appropriate 2 modified fredholm determinant we give a simple proof of convergence in location and multiplicity of hill s method for numerical approximation of spectra of periodic coefficient ordinary differential operators our results apply to operators of nondegenerate type under the condition that the principal coefficient matrix be symmetric positive definite automatically satisfied in the scalar case notably this includes a large class of non self adjoint operators which previously had not been treated in a simple way the case of general coefficients depends on an interesting operator theoretic question regarding properties of toeplitz matrices

loss of stability at turning points in nonlinear elliptic variational inequalities a considerable amount of work has been devoted in recent years to stationary free boundary problems in particular obstacle problems have been the subject of extensive work concerning existence estimates and regularity of the solution or of the coincidence set and occurrence of multiple solutions in 6 8 161 the authors consider obstacle problems with nonlinear forcing terms leading to multiple solutions hence to turning points when a real parameter appears in the forcing term a local study of the branches of solutions is set up the results obtained are the extension in a weaker form of those well known for equations ll 12 however some important classical questions remain open for obstacle problems

approximating eigenfunctions of fredholm operators in banach spaces abstract a singular fredholm operator a is perturbed by an operator of finite rank to obtain an invertible operator b theory previously developed for a and b in hilbert spaces is extended here to banach spaces the operator b 1 is used to construct independent elements in the null spaces n a n a2 n ak for some positive integer k and a basis for n a and n a2 the theory is used to compute approximations to eigenfunctions and generalized eigenfunctions of integral operators

approximation of the spectrum of closed operators the determination of normal modes of a rotating basin note univ michigan dept math ann arbor mi 48109 ecole polytech ctr math appl f 91128 palaiseau france descloux j ecole prat hautes etud dept math f 75231 paris 05 france isi document delivery no le068times cited 3cited reference count 17 reference asn article 1981 004doi 10 2307 2007731 record created on 2006 08 24 modified on 2017 05 12

on unipotent supports of reductive groups with a disconnected centre let g be a simple algebraic group defined over a finite field of good characteristic with associated frobenius endomorphism f in this article we extend an observation of lusztig which gives a numerical relationship between an ordinary character of gf and its unipotent support to the case where z g is disconnected we then use this observation in some applications to the ordinary character theory of gf 2013 elsevier inc

local piecewise polynomial projection methods for an o d e which give high order convergence at knots local projection methods which yield c sup m 1 piecewise polynomials of order m k as approximate solutions of a boundary value problem for an mth order ordinary differential equation are determined by the k linear functionals at which the residual error in each partition interval is required to vanish on we develop a condition on these k functionals which implies breakpoint superconvergence of derivatives of order less than m for the approximating piecewise polynomials the same order of superconvergence is associated with eigenvalue problems a discrete connection between two particular projectors yielding o vertical bardvertical bar sup 2k superconvergence namely a collocation at the k gauss legendre points in each partition interval and b essential least squares i e local moment methods is made by asking that this same order of superconvergence result when using collocation at k r points per interval and simultaneous local orthogonality of the residual to polynomials of order r the k r points then necessarily form a subset of the k gauss legendre points

instability of periodic traveling wave solutions in a modified fitzhugh nagumo model for excitable media we introduce a two variable system of reaction diffusion equations for excitable media we numerically investigate the existence and stability of periodic traveling wave solutions in a two dimensional parameter plane our results based on the method of continuation show a stability change of eckhaus type there are two families of periodic traveling waves fast and slow the fast family is stable in the case of standard fitzhugh nagumo excitable system however we observe that the fast family becomes unstable in our model consequently it bifurcates to an oscillating wave we explain this phenomenon by numerically calculating the essential spectra of the periodic traveling wave solutions moreover we study the stability of the periodic traveling wave solutions for the aliev panfilov excitable system and compare its results with the proposed model we show that the stability boundary of our model agrees qualitatively with that of the aliev panfilov model whereas it is different in the fitzhugh nagumo model in addition our results based on the stability boundary of the models show oscillating periodic traveling waves except for the fitzhugh nagumo model

cuspidal characters and automorphisms abstract we investigate the action of outer automorphisms of finite groups of lie type on their irreducible characters we obtain a definite result for cuspidal characters as an application we verify the inductive mckay condition for some further infinite families of simple groups at certain primes

on the spectra of periodic waves for infinite dimensional hamiltonian systems abstract we consider the problem of determining the spectrum for the linearization of an infinite dimensional hamiltonian system about a spatially periodic traveling wave by using a bloch wave decomposition we recast the problem as determining the point spectra for a family of operators j l where j is skew symmetric with bounded inverse and l is symmetric with compact inverse our main result relates the number of unstable eigenvalues of the operator j l to the number of negative eigenvalues of the symmetric operator l the compactness of the resolvent operators allows us to greatly simplify the proofs as compared to those where similar results are obtained for linearizations about localized waves the theoretical results are general and apply to a larger class of problems than those considered herein the theory is applied to a study of the spectra associated with periodic and quasi periodic solutions to the nonlinear schrodinger equation as well as periodic solutions to the generalized korteweg de vries equation with power nonlinearity

pattern solutions of the klausmeier model for banded vegetation in semiarid environments iv slowly moving patterns and their stability banded vegetation is a characteristic feature of semiarid environments comprising stripes of vegetation running parallel to the contours on hillsides separated by stripes of bare ground mathematical modelling plays an important role in the study of this phenomenon and the klausmeier model is one of the oldest and most established consisting of coupled reaction diffusion advection equations for plant biomass and water density in this model the dimensionless parameter corresponding to slope gradient is much larger than the other parameters and this paper is part of a series investigating the asymptotic form of pattern solutions for large values of the slope parameter the pattern solutions move uphill with a constant speed c say and the focus of this paper is c o 1 i begin by deriving the leading order form of the patterns and the region of parameter space in which they occur for c o 1 using this i show that all patterns with c o 1 are unstable as model solutions for sufficiently la

local decay of waves on asymptotically flat stationary space times author s tataru daniel abstract in this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t 3 local uniform decay rate for linear waves this work was motivated by open problems concerning decay rates for linear waves on schwarzschild and kerr backgrounds where such a decay rate has been conjectured by r price our results apply to both of these cases

whitham modulation theory for the kadomtsev petviashvili equation the genus 1 kadomtsev petviashvili kp whitham system is derived for both variants of the kp equation namely the kpi and kpii equations the basic properties of the kp whitham system including symmetries exact reductions and its possible complete integrability together with the appropriate generalization of the one dimensional riemann problem for the korteweg de vries equation are discussed finally the kp whitham system is used to study the linear stability properties of the genus 1 solutions of the kpi and kpii equations it is shown that all genus 1 solutions of kpi are linearly unstable while all genus 1 solutions of kpii are linearly stable within the context of whitham theory

metastability of solitary roll wave solutions of the st venant equations with viscosity we study by a combination of numerical and analytical evans function techniques the stability of solitary wave solutions of the st venant equations for viscous shallowwater ow down an incline and related models our main result is to exhibit examples of metastable solitary waves for the st venant equations with stable point spectrum indicating coherence of the wave prole but unstable essential spectrum indicating oscillatory convective instabilities shed in its wake we propose a mechanism based on dynamic spectrum of the wave prole by which a wave train of solitary pulses can stabilize each other by de amplication of convective instabilities as they pass through successive waves we present numerical time evolution studies supporting these conclusions which bear also on the possibility of stable periodic solutions close to the homoclinic for the closely related viscous jin xin model by contrast for which the essential spectrum is stable we show using the stability index of gardner zumbrun that solitary wave pulses are always exponentially unstable possessing point spectra with positive real part

parametric resonance in immersed elastic boundaries in this paper we investigate the stability of a fluid structure interaction problem in which a flexible elastic membrane immersed in a fluid is excited via periodic variations in the elastic stiffness parameter this model can be viewed as a prototype for active biological tissues such as the basilar membrane in the inner ear or heart muscle fibers immersed in blood problems such as this in which the system is subjected to internal forcing through a parameter can give rise to parametric resonance we formulate the equations of motion in two dimensions using the immersed boundary formulation assuming small amplitude motions we can apply floquet theory to the linearized equations and derive an eigenvalue problem whose solution defines the marginal stability boundaries in parameter space the eigenvalue equation is solved numerically to determine values of fiber stiffness and fluid viscosity for which the problem is linearly unstable we present direct numerical simulations of the fluid structure interaction problem using the immersed boundary method that verify the existence of the parametric resonances suggested by our analysis

numerical analysis of periodic solutions and bifurcations in the planetary annulus problem this paper discusses the dynamics of particles orbiting planetary rings under a general central potential starting with the mathematical description of the dynamical system we analyze the motion of a particle with infinitesimal mass as attracted by a central body surrounded by a homogeneous circular annular disk throughout the paper we carry out an analytic search of the most relevant equilibria solutions and based on that we investigate numerically the stability matrix of the system to find stability inequalities in this way we describe the in plane and out of plane motion by means of the numerical continuation of a wide number of uni parametric families of planar and spatial periodic orbits we present a description of the main families of periodic orbits encountered their bifurcations and linear stability with the aim of reproducing a more realistic scenario we analyse different mass proportions between the annulus and the central body we consider an oblate planet and we also include a composition of rings in the dynamical model

numerical continuation methods for studying periodic travelling wave wavetrain solutions of partial differential equations abstract periodic travelling waves wavetrains are an important solution type for many partial differential equations in this paper i review the use of numerical continuation for studying these solutions i discuss the calculation of the form and stability of a given periodic travelling wave and the calculation of boundaries in a two dimensional parameter plane for wave existence and stability i also describe the automated implementation of these numerical continuation procedures via the software package wavetrain http www ma hw ac uk wavetrain i conclude by discussing ongoing work on numerical continuation methods for determining the absolute stability of periodic travelling waves

computing spectra of linear operators using the floquet fourier hill method in order to establish the stability of an equilibrium solution u of an infinite dimensional dynamical system u x u one is interested in the spectrum of the linear operator lu obtained by linearizing the dynamical system around u we use a spectrally accurate method for the computation of the spectrum of the maximal extension of the operator lu the method is particularly well suited to the case of periodic u although no periodic boundary conditions on the perturbations are imposed by incorporating the fundamentals of floquet theory an almost uniform approximation to the entire spectrum of the maximal extension is obtained as opposed to an approximation of a few selected elements the numerical component of the method is limited to i choosing the size of the matrices to be used and ii an eigenvalue solver such as the qr algorithm compared to often used finite difference approaches the method is an order of magnitude faster for comparable accuracy we illustrate that the method is efficiently extended to problems defined on the whole line

numerical inverse scattering for the korteweg de vries and modified korteweg de vries equations abstract recent advances in the numerical solution of riemann hilbert problems allow for the implementation of a cauchy initial value problem solver for the korteweg de vries equation kdv and the defocusing modified korteweg de vries equation mkdv without any boundary approximation borrowing ideas from the method of nonlinear steepest descent this method is demonstrated to be asymptotically accurate the method is straightforward for the case of defocusing mkdv due to the lack of poles in the riemann hilbert problem and the boundedness properties of the reflection coefficient solving kdv requires the introduction of poles in the riemann hilbert problem and more complicated deformations the introduction of a new deformation for kdv allows for the stable asymptotic computation of the solution in the entire spacial and temporal plane kdv and mkdv are dispersive equations and this method can fully capture the dispersion with spectral accuracy thus this method can be used as a benchmarking tool for determining the effectiveness of future numerical methods designed to capture dispersion this method can easily be adapted to other integrable equations with riemann hilbert formulations such as the nonlinear schrodinger equation

absolute and convective instabilities of waves on unbounded and large bounded domains abstract instabilities of nonlinear waves on unbounded domains manifest themselves in different ways an absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain in contrast convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time perturbations decay locally at each given point in the unbounded domain wave packets are convected towards infinity in experiments as well as in numerical simulations bounded domains are often more relevant we are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the aforementioned in stability properties of a wave these effects depend upon the boundary conditions that are imposed on the bounded domain we compare the spectra of the linearized evolution operators on unbounded and bounded domains for two classes of boundary conditions it is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately spectra for separated boundary conditions behave in quite a different way firstly separated boundary conditions may generate additional isolated eigenvalues secondly the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain instead the so called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights we interpret the difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain in particular it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave but that convectively unstable waves can destabilize under domain truncation the theoretical predictions are compared with numerical computations

basic reproduction ratios for almost periodic compartmental models with time delay abstract the theory of basic reproduction ratio r 0 is established for a large class of almost periodic and time delayed compartmental population models we first present some dynamical properties for linear almost periodic functional differential systems by using the product space and evolution semigroup approach we then prove that r 0 1 has the same sign as the exponential growth bound of an associated linear system as an application we apply the developed theory to an almost periodic seir model with an incubation period and obtain a threshold result on its global dynamics in terms of r 0 finally we present some numerical simulations numerical simulations indicate that prolonging the length of incubation period is beneficial for the control of the disease in addition a simple model shows that the basic reproduction ratio may be underestimated or overestimated if an almost periodic coefficient is approximated by a periodic one

elliptic solutions of the defocusing nls equation are stable the stability of the stationary periodic solutions of the integrable one dimensional cubic defocusing nonlinear schrodinger nls equation is reasonably well understood especially for solutions of small amplitude in this paper we exploit the integrability of the nls equation to establish the spectral stability of all such stationary solutions this time by explicitly computing the spectrum and the corresponding eigenfunctions associated with their linear stability problem an additional argument using an appropriate krein signature allows us to conclude the nonlinear orbital stability of all stationary solutions of the defocusing nls equation with respect to so called subharmonic perturbations perturbations that have period equal to an integer multiple of the period of the amplitude of the solution all results presented here are independent of the size of the amplitude of the solutions and apply equally to solutions with trivial and nontrivial phase profiles

sharp low frequency resolvent estimates on asymptotically conical manifolds on a class of asymptotically conical manifolds we prove two types of low frequency estimates for the resolvent of the laplace beltrami operator the first result is a uniform l 2 rightarrow l 2 bound for scal r 1 delta g z 1 scal r 1 when mbox re z is small with the optimal weight scal r 1 the second one is about powers of the resolvent for any integer n we prove uniform l 2 rightarrow l 2 bounds for scal epsilon r n epsilon 2 delta g z n scal epsilon r n when mbox re z belongs to a compact subset of 0 infty and 0 epsilon ll 1 these results are obtained by proving similar estimates on a pure cone with a long range perturbation of the metric at infinity

global threshold dynamics of an infection age structured sir epidemic model with diffusion under the dirichlet boundary condition abstract in this paper we are concerned with the global asymptotic behavior of an infection age structured sir epidemic model with diffusion in a general n dimensional bounded spatial domain under the homogeneous dirichlet boundary condition by using the method of characteristics we reformulate the model into a system of a reaction diffusion equation and a volterra integral equation we define the basic reproduction number r 0 by the spectral radius of a compact positive linear operator and show that if r 0 1 then the disease free steady state is globally attractive whereas if r 0 1 then a positive endemic steady state exists and the system is uniformly persistent by numerical simulation for the 2 dimensional case we show that r 0 depends on the shape of the spatial domain this result is in contrast with the case of the homogeneous neumann boundary condition in which r 0 is independent of the spatial domain

totally orthogonal finite simple groups we prove that if g is a finite simple group then all irreducible complex representations of g may be realized over the real numbers if and only if every element of g may be written as a product of two involutions in g this follows from our result that if q is a power of 2 then all irreducible complex representations of the orthogonal groups mathrm o pm 2n mathbb f q may be realized over the real numbers we also obtain generating functions for the sums of degrees of several sets of unipotent characters of finite orthogonal groups

dynamics in a predator prey model with space and noise in this paper we presented a predator prey model with spatial diffusion and density dependent noise it was found that the spatial model without noise has spotted pattern however combined with density dependent noise the predator prey model exhibits transition from spotted to labyrinth pattern furthermore when noise intensity and temporal correlation are in appropriate levels the populations will extinct the obtained results suggest that stochastic factors may play an important role in populations dynamics which well enrich the findings in predator prey models

an equivariant bijection between irreducible brauer characters and weights for sp 2n q abstract the longstanding alperin weight conjecture and its blockwise version have been reduced to simple groups recently by navarro tiep spath and koshitani thus to prove this conjecture it suffices to verify the corresponding inductive condition for all finite simple groups the first is to establish an equivariant bijection between irreducible brauer characters and weights for the universal covering groups of simple groups assume q is a power of some odd prime p we first prove the blockwise alperin weight conjecture for sp 2 n q and odd non defining characteristics if the decomposition matrix of sp 2 n q is unitriangular with respect to an aut sp 2 n q stable basic set this assumption holds for linear primes we can establish an equivariant bijection between the irreducible brauer characters and weights

equivariance and extendibility in finite reductive groups with connected center we show that several character correspondences for finite reductive groups g are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to g has connected center the correspondences we consider are the so called jordan decomposition of characters introduced by lusztig and the generalized harish chandra theory of unipotent characters due to broue malle michel in addition we consider a correspondence giving character extensions due to the second author in order to verify the inductive mckay condition from isaacs malle navarro for the non abelian finite simple groups of lie types 3 mathsf d 4 mathsf e 8 mathsf f 4 2 mathsf f 4 and mathsf g 2

a unipotent support for irreducible representations 1 1 let g f be the group of fq rational points of a connected reductive algebraic group g defined over a finite field f throughout this paper we shall assume that the characteristic of the ground field is large the following is part of problem ii in l9 let p be an irreducible complex character of g f show that there is a unique umpotent class c in g which has the property that cga c cfqi p g 0 and has maximal dimension among classes with this property one of the results of this paper is a solution of this problem we also prove the following refinement if g e gf is such that tr g p 0 then the unipotent part of g lies in c or in a conjugacy class of dimension dim c we call c the unipotent support of p another map from the set of irreducible representations of gf to unipotent classes in g has been defined in l8 13 43 this was the composition of two maps one from irreducible representations of gf to a set of parameters in the langlands dual of g and one from this set of parameters to unipotent classes in g another conjectural definition of this map which did not pass through the langlands dual has been proposed by kawanaka kl k2 k3 in terms of generalized gelfand graev representations in analogy with real lie groups he uses the name wave front set for the unipotent class attached to a representation one of the results of this paper is a proof see section 11 of kawanaka s conjecture that his wave front set is well defined and that it coincides with the one defined in l8 this has been known earlier from the work of kawanaka in certain special cases we also show that the unipotent support of an irreducible representation coincides with the wave front set of the dual representation

approximations of strongly continuous families of unbounded self adjoint operators the problem of approximating the discrete spectra of families of self adjoint operators that are merely strongly continuous is addressed it is well known that the spectrum need not vary continuously as a set under strong perturbations however it is shown that under an additional compactness assumption the spectrum does vary continuously and a family of symmetric finite dimensional approximations is constructed an important feature of these approximations is that they are valid for the entire family uniformly an application of this result to the study of plasma instabilities is illustrated

on the convergence of hill s method hill s method is a means to numerically approximate spectra of linear differential operators with periodic coefficients in this paper we address different issues related to the convergence of hill s method we show the method does not produce any spurious approximations and that for selfadjoint operators the method converges in a restricted sense furthermore assuming convergence of an eigenvalue we prove convergence of the associated eigenfunction approximation in the l 2 norm these results are not restricted to selfadjoint operators finally for certain selfadjoint operators we prove that the rate of convergence of hill s method to the least eigenvalue is faster than any polynomial power

wave train selection behind invasion fronts in reaction diffusion predator prey models abstract wave trains or periodic travelling waves can evolve behind invasion fronts in oscillatory reaction diffusion models for predator prey systems although there is a one parameter family of possible wave train solutions in a particular predator invasion a single member of this family is selected sherratt 1998 13 has predicted this wave train selection using a system that is a valid approximation near a supercritical hopf bifurcation in the corresponding kinetics and when the predator and prey diffusion coefficients are nearly equal away from a hopf bifurcation or if the diffusion coefficients differ somewhat these predictions lose accuracy we develop a more general wave train selection prediction for a two component reaction diffusion predator prey system that depends on linearizations at the unstable homogeneous steady states involved in the invasion front this prediction retains accuracy farther away from a hopf bifurcation and can also be applied when the predator and prey diffusion coefficients are unequal we illustrate the selection prediction with its application to three models of predator invasions

on stability of generalized short crested water waves abstract in this paper we take up the question of dynamic stability of genuinely two dimensional generalized hexagonal traveling wave patterns on the surface of a three dimensional ideal fluid i e stability of generalized short crested wave gscw solutions of the water wave problem we restrict ourselves to a study of spectral stability which considers the linearization of the water wave operator about one of these traveling generalized hexagonal waves and draws conclusions about stability from the spectral data of the resulting linear operator within the class of perturbations we are allowed to study for a wide range of geometrical parameters we find stable traveling waveforms which eventually destabilize with features that depend strongly on the problem configuration in particular we find zones of instability for patterns shaped as symmetric diamonds while such zones are absent for asymmetric configurations furthermore we note that within a geometrical configuration as a generalized scw ratio is varied essentially the character of the linear solution these waves become more unstable as the waves become more asymmetric

the spectral approximation of linear operators with applications to the computation of eigenelements of differential and integral operators we are concerned with the numerical solution of the eigenvalue problem t varphi lambda varphi varphi ne 0 where t is a linear operator in a bana ch space t may represent a bounded integral operator or a closed differential operator with bounded inverse the linear operator t and its approximation t n are defined in the same space perturbation theory is then a suitable framework for our problem we present in the first part a systematic study of the various notions of convergence t n to t which imply the convergence of the eigenvalues with preservation of the multiplicities the results are then applied to practical methods for integral and differential operators in the second part we present convergence rates analytic perturbation theory is used to refine on the computed eigenelements of an integral operator and to produce localization results on the eigenelements finally superconvergence results are discussed both for integral and differential operators

on the stability of periodic travelling waves with large spatial period abstract in many circumstances a pulse to a partial differential equation pde on the real line is accompanied by periodic wave trains that have arbitrarily large period it is then interesting to investigate the pde stability of the periodic wave trains given that the pulse is stable using the evans function gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves in this article the precise location of these circles is determined it is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse as a consequence periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi hump pulses are created that is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations the proof uses lyapunov schmidt reduction and relies on the existence of exponential dichotomies the approach is also applicable to periodic waves with large spatial period of elliptic problems on r n or on unbounded cylinders r with bounded

spectral analysis of n body schrodinger operators for a large class of two body potentials we solve two of the main problems in the spectral analysis of multiparticle quantum hamiltonians explicitly we prove that the point spectrum lies in a closed countable set and describe that set in terms of the eigenvalues of hamiltonians of subsystems and that there is no singular continuous spectrum we accomplish this by extending mourre s work on three body problems to n body problems

the stability spectrum for elliptic solutions to the focusing nls equation abstract we present an analysis of the stability spectrum of all stationary elliptic type solutions to the focusing nonlinear schrodinger equation nls an analytical expression for the spectrum is given from this expression various quantitative and qualitative results about the spectrum are derived specifically the solution parameter space is shown to be split into four regions of distinct qualitative behavior of the spectrum additional results on the stability of solutions with respect to perturbations of an integer multiple of the period are given as well as a procedure for approximating the greatest real part of the spectrum

computing absolute and essential spectra using continuation a continuation approach to the computation of essential and absolute spectra of dierential operators on the real line is presented the advantage of this approach compared with direct eigenvalue computations for the discretized operator are the ecient and accurate computation of selected parts of the spectrum typically those near the imaginary axis and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability we also discuss the implementation and usage of this approach with the software package auto and provide example computations for the fitzhugh nagumo and the complex ginzburg landau equation

a nonlocal and time delayed reaction diffusion eco epidemiological predator prey model abstract in this paper we propose and investigate a nonlocal and time delayed reaction diffusion predator prey model with disease spread in the prey which incorporates spatial heterogeneity and assimilation lag of consumed prey biomass into predator biomass the si type subsystem of prey without predator is first studied and we establish the threshold dynamics in terms of the basic reproduction number based on epidemiology then we show a series of sufficient conditions for the co existence of prey and predator species the global extinction of predator species and the global extinction of both the prey and predator species finally a special example is explored via numerical simulations where the influences of nonlocal effect spatial heterogeneity and time delay on the invasion of the predator species are probed

on the spectral stability of solitary wave solutions of the vector nonlinear schrodinger equation we consider a system of coupled cubic nonlinear schrodinger nls equations where the interaction coefficients jk are real the spectral stability of solitary wave solutions both bright and dark is examined both analytically and numerically our results build on preceding work by nguyen et al and others specifically we present closed form solitary wave solutions with trivial and nontrivial phase profiles their spectral stability is examined analytically by determining the locus of their essential spectrum their full stability spectrum is computed numerically using a large period limit of hill s method we find that all nontrivial phase solutions are unstable while some trivial phase solutions are spectrally stable to our knowledge this paper presents the first investigation of the stability of the solitary waves of the coupled cubic nls equation without the restriction that all components j are proportional to sech

periodic finite genus solutions of the kdv equation are orbitally stable abstract the stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade the kdv equation is known to have large families of periodic solutions that are parameterized by hyperelliptic riemann surfaces they are generalizations of the famous multi soliton solutions we show that all such periodic solutions are orbitally stable with respect to subharmonic perturbations perturbations that are periodic with period equal to an integer multiple of the period of the underlying solution

on the inductive blockwise alperin weight condition for classical groups abstract recently there has been substantial progress on the alperin weight conjecture as a step to establish the alperin weight conjecture for all finite groups we prove the inductive blockwise alperin weight condition for simple groups of classical type under some additional assumption

the effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction diffusion systems abstract many oscillatory biological systems show periodic travelling waves these are often modelled using coupled reaction diffusion equations however the effects of different movement rates diffusion coefficients of the interacting components on the predictions of these equations are largely unknown here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed time period wavelength amplitude and stability of periodic wave solutions we focus on two sets of kinetics that are commonly used in ecological applications lambda omega equations which are the normal form of an oscillatory coupled reaction diffusion system close to a supercritical hopf bifurcation and a standard predator prey model our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one parameter family of periodic travelling wave solutions the position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients in all cases stability changes through an eckhaus sideband instability these effects are always symmetrical in the two diffusion coefficients for the lambda omega equations but are asymmetric in the predator prey equations especially when the limit cycle of the kinetics is of large amplitude in particular there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator prey scenario our results also show the existence of a one parameter family of travelling waves but not necessarily a hopf bifurcation for all values of the diffusion coefficients simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms and our analysis of the travelling wave families assists in the understanding of these effects

stability analysis of 1 1 dimensional cnoidal waves in media with cubic nonlinearity in the present paper we perform stability analysis of stationary 1 1 dimensional cnoidal waves of cn and dn types anomalous group velocity dispersion and sn type normal group velocity dispersion the mathematical model is based on the nonlinear schrodinger equation with this aim we developed a method that takes into consideration the properties of complex eigenvalues of cauchy matrix for perturbation vectors we show that cnoidal sn wave is stable in the whole domain of its existence whereas cn and dn waves are unstable the instability of cn and dn waves is suppressed in the limiting case of strong localization when waves evolve into a set of well separated fundamental bright solitons

a nonlocal and time delayed reaction diffusion model of dengue transmission a nonlocal and time delayed reaction diffusion model of dengue fever is first proposed that incorporates the aquatic stage the winged stage and the incubation periods of the dengue virus within mosquitos and hosts then the basic reproduction number r 0 is established for the model system and an explicit formula of r 0 is obtained in the case of spatially homogeneous infections it is shown that this r 0 gives the threshold dynamics in the sense that the disease free equilibrium is asymptotically stable if r 0 1 the influences of diffusion coefficients time delays and infection heterogeneity on the spread of the disease are also studied via numerical simulations it turns out that the infection risk may be underestimated if the spatially averaged parameters are used to compute the basic reproduction number for spatially heterogeneous infections

spectral properties of diagonally dominant infinite matrices part i on etablit des theoremes de type gersgorin pour un operateur matriciel infini non borne diagonalement dominant a agissant sur l p pour 1 p

large scale eigenvector approximation via hilbert space embedding nystrom the nystrom method approximates eigenvectors of a given kernel matrix by randomly sampling subset of data previous researches focus on good kernel approximation while the quality of eigenvector approximation is rarely explored in online eigenvector approximation method one can minimize the kernel approximation error to guarantee a good eigenvector approximation however in this work we paradoxically prove that for batch approximation methods like nystrom it is no longer true this unexpected discovery opens a question what criterion should we use in nystrom to generate a decent eigenvector approximation to address this problem we propose a novel criterion named hilbert space embedding hse nystrom criterion which directly minimizes the eigenvector approximation error the proposed hse criterion provides a general framework to approximate eigenvectors within linear time and space complexity we then show that we can rediscover many successful nystrom methods with the proposed criterion including k means nystrom and density nystrom to further demonstrate the power of our criterion we actually design a novel algorithm to approximate eigenvectors of laplacian matrices based on the proposed criterion with better accuracy among existing linear complexity methods we demonstrate the efficiency and efficacy of our proposal in numerical experiments highlightsa new criterion for nystrom method to find accurate eigenvector approximation the conventional criterion does not work well in eigenvector approximation we need to adjust kernel width together with average weight adaptively propose a new criterion to adaptively optimize the parameters efficiently

pattern solutions of the klausmeier model for banded vegetation in semi arid environments iii the transition between homoclinic solutions abstract self organised patterns of vegetation are a characteristic feature of semi deserts on hillsides these typically comprise vegetation bands running parallel to the contours separated by regions of bare ground tiger bush the present study concerns the klausmeier mathematical model for this phenomenon c a klausmeier regular and irregular patterns in semiarid vegetation science 284 1999 1826 1828 which is one of the earliest and most influential of the various theoretical models for banded vegetation the model is a system of reaction diffusion advection equations and after rescaling it contains three dimensionless parameters one of which the slope parameter is much larger than the other two the present study is the third in a series of papers in which the author exploits the large value of the slope parameter to obtain leading order approximations to the parameter regions in which patterns exist and to the form of these patterns the boundary of the parameter region giving patterns consists in part of two loci of homoclinic solutions that are homoclinic to different steady states the present paper concerns behaviour for parameters close to the intersection point of these loci the author shows that this part of parameter space divides naturally into three regions with a different solution structure in each in one region the solution corresponds to a limit cycle of a reduced system of ordinary differential equations the other two regions involve multiple matched layers as part of the analysis the author derives formulae for the homoclinic solution loci and for the location of their intersection all of the results are valid to leading order for large values of the slope parameter the author presents a detailed numerical verification of his analytical results the paper concludes with discussions of the ecological implications of the results and the main outstanding mathematical questions