the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

asymptotic behavior near planar transition fronts for the cahn hilliard equation abstract we consider the asymptotic behavior of perturbations of planar wave solutions arising in the cahn hilliard equation in space dimensions d 2 such equations are well known to arise in the study of spinodal decomposition a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into regions in which one component or the other is dominant with these regions separated by steep transition layers a critical feature of the cahn hilliard equation in one space dimension is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability the analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator leading in the case of stability moves into the negative real half plane with cubic scaling 3 where r d 1 denotes a fourier variable associated with spatial components transverse to the planar wave under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from planar wave solutions establishing asymptotic stability for initial perturbations decaying at an appropriate algebraic rate in an l 1 norm of the transverse variables

on decomposition matrices for graded algebras abstract we study the relations between the decomposition matrix of a module over a graded algebra and the decomposition matrix of its restriction to the grading subalgebra we show that under appropriate hypotheses if one of the decomposition matrices is stable unitriangular then so is the other a particular case is when one of them is square unitriangular then both are using these results we describe the decomposition matrices of the hecke algebras of the complex reflection groups g de e n with respect to the ones of the ariki koike algebras or the decomposition matrices of q schur algebras of type dn with respect to the ones of q schur algebras of type bn

hopf bifurcation of viscous shock waves in compressible gas dynamics and mhd extending our previous results for artificial viscosity systems we show under suitable spectral hypotheses that shock wave solutions of compressible navier stokes and magnetohydrodynamics equations undergo hopf bifurcation to nearby time periodic solutions the main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in h s for data in h s we accomplish this by a novel energy estimate carried out in lagrangian coordinates interestingly this estimate is false in eulerian coordinates at the same time we greatly sharpen and simplify the analysis of the previous work

transition to longitudinal instability of detonation waves is generically associated with hopf bifurcation to time periodic galloping solutions we show that transition to longitudinal instability of strong detonation solutions of reactive compressible navier stokes equations is generically associated with hopf bifurcation to nearby time periodic galloping or pulsating solutions in agreement with physical and numerical observation in the process we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated evans function and obtain the first complete nonlinear stability result for strong detonations of the reacting navier stokes equations in the limit as amplitude hence also heat release goes to zero the analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works

on the inductive alperin mckay and alperin weight conjecture for groups with abelian sylow subgroups abstract we study the inductive alperin mckay conjecture the inductive isaacs navarro refinement and the inductive blockwise alperin weight conjecture for groups of lie type in the generic case of abelian sylow l subgroups we also show that the alternating groups the suzuki groups and the ree groups satisfy the inductive condition necessary for spath s reduction of the blockwise alperin weight conjecture to the case of simple groups

high frequency asymptotics and one dimensional stability of zel dovich von neumann doring detonations in the small heat release and high overdrive limits we establish one dimensional spectral or normal modes stability of zel dovich von neumann doring detonations in the small heat release limit and the related high overdrive limit with heat release and activation energy held fixed verifying numerical observations made by erpenbeck in the 1960s the key technical points are a strategic rescaling of parameters converting the infinite overdrive limit to a finite regular perturbation problem and a careful high frequency analysis depending uniformly on model parameters the latter recovers and extends to arbitrary amplitudes the important result of high frequency stability established by erpenbeck by somewhat different techniques notably the techniques used here yield quantitative estimates well suited for numerical stability investigation

asymptotic behavior near transition fronts for equations of generalized cahn hilliard form we consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary pde of generalized cahn hilliard form in one space dimension such equations are well known to arise in the study of spinodal decomposition a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into its two components with their concentrations separated by sharp transition layers motivated by work of bricmont kupiainen and taskinen 5 we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions a critical feature of the cahn hilliard equation is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from standing wave solutions establishing phase asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate

quantum weyl reciprocity and tilting modules quantum weyl reciprocity relates the representation theory of hecke algebras of type a with that of q schur algebras this paper establishes that weyl reciprocity holds integrally i e over the ring q q 1 of laurent polynomials and that it behaves well under base change a key ingredient in our approach involves the theory of tilting modules for q schur algebras new results obtained in that direction include an explicit determination of the ringel dual algebra of a q schur algebra in all cases in particular in the most interesting situation the ringel dual identifies with a natural quotient algebra of the hecke algebra

on the decomposition numbers of the finite general linear groups ii let g gl q q a prime power and let r be an odd prime not dividing q let s be a semisimple element of g of order prime to r and assume that r divides qdeg x 1 for all elementary divisors a of s relating representations of certain hecke algebras over symmetric groups with those of g we derive a full classification of all modular irreducible modules in the r block bh of g with semisimple part s the decomposition matrix d of bs may be partly described in terms of the decomposition matrices of the symmetric groups corresponding to the hecke algebras above moreover d is lower unitriangular this applies in particular to all r blocks of g if r divides q 1 thus in this case the r decomposition matrix of g is lower unitriangular introduction the modular representation theory of finite groups of lie type defined over a field of characteristic p is naturally divided into the cases of equal characteristic r p and unequal characteristic r p this paper begins the study of decomposition numbers in the unequal characteristic case of the finite general linear groups when r 2 let g be the full linear group of degree n over gf q and 2 r be a prime not dividing q let s c g be semisimple and consider the geometric conjugacy class s g of irreducible characters corresponding to the g conjugacy class of s using basic properties of the deligne lusztig operators introduced in 7 we will determine a parabolic subgroup p ps of g and a cuspidal irreducible character of x of p such that s g is just the set of constituents of xg the induced character let r r k be an r modular system for g and m a kp module affording x then end kg m is called the hecke algebra of m compare 5 and has been calculated in a more general context by r b howlett and g i lehrer in 12 it turns out that end kg m k w the hecke algebra of w where w wj denotes the weyl group of cg s the centralizer of s in g compare 1 2 4 13 in particular k w has a basis tw iw c w such that the structure constants of the multiplication are all contained in r c k so let r w be the r order in k w generated by wtiw ew and r w r r r w choosing a special rp lattice s in m and using the classification of the r blocks of rg given by p fong and b srinivasan in received by the editors july 25 1983 and in revised form october 18 1983 may 23 1984 and september 1 1984 1980 mathematics subject classification primary 20c20 secondary 20g40 this work was supported by the deutsche forschungsgemeinschaft 1985 american mathematical society 0002 9947 85 1 00 25 per page

an automatic orthonormalization method for solving stiff boundary value problems abstract a new initial value method is described based on a remark by drury for solving stiff linear differential two point cigenvalue and boundary value problems the method is extremely reliable it is especially suitable for high order differential systems and it is capable of accommodating realms of stiffness which other methods cannot reach the key idea behind the method is to decompose the stiff differential operator into two non stiff operators one of which is nonlinear the nonlinear one is specially chosen so that it advances an orthonormal frame indeed the method is essentially a kind of automatic orthonormalization the second is auxiliary but it is needed to determine the required function the usefulness of the method is demonstrated by calculating some eigenfunctions for an orr sommerfeld problem when the reynolds number is as large as 10

existence and stability of viscoelastic shock profiles we investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by antman and malek madani we establish that the resulting equations fall into the class of symmetrizable hyperbolic parabolic systems hence spectral stability implies linearized and nonlinear stability with sharp rates of decay the new contributions are treatment of the compressible case formulation of a rigorous nonlinear stability theory including verification of stability of small amplitude lax shocks and the systematic incorporation in our investigations of numerical evans function computations determining stability of large amplitude and nonclassical type shock profiles

the compound matrix method for ordinary differential systems abstract a generalization of the compound matrix method is presented to deal with eigenvalue and boundary value problems involving unstable systems of ordinary differential equations details are given for fourth and sixth order problems next is shown that a simple equivalence relation exists between the compounds of the solution matrices of the eigenvalue problems and their adjoints and how this relation can be exploited to simplify the calculation of the adjoint eigenfunctions is discussed using the orr sommerfeld problem as an example it is also shown how the techniques used in the derivation of certain auxiliary systems which play a crucial role in the generalization of the compound matrix method can also provide an alternate method for the direct computation of certain quantities defined in terms of the eigenfunctions finally there is a brief discussion of the relationship between the compound matrix method anf the riccati method

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

convex entropy hopf bifurcation and viscous and inviscid shock stability we consider by a combination of analytical and numerical techniques some basic questions regarding the relations between inviscid and viscous stability and the existence of a convex entropy specifically for a system possessing a convex entropy in particular for the equations of gas dynamics with a convex equation of state we ask 1 can inviscid instability occur 2 can viscous instability not detected by inviscid theory occur 3 can there occur the necessarily viscous effect of hopf bifurcation or galloping instability and perhaps most important from a practical point of view 4 as shock amplitude is increased from the stable weak amplitude limit can there occur a first transition from viscous stability to instability that is not detected by inviscid theory we show that 1 does occur for strictly hyperbolic genuinely nonlinear gas dynamics with certain convex equations of state while 2 and 3 do occur for an artifically constructed system with convex viscosity compatible entropy we do not know of an example for which 4 occurs leaving this as a key open question in viscous shock theory related to the principal eigenvalue property of sturm liouville and related operators in analogy with and partly proceeding close to the analysis of smith on non uniqueness of the riemann problem we obtain convenient criteria for shock in stability in the form of necessary and sufficient conditions on the equation of state

grassmannian spectral shooting we present a new numerical method for computing the pure point spectrum associated with the linear stability of coherent structures in the context of the evans function shooting and matching approach all the relevant information is carried by the flow projected onto the underlying grassmann manifold we show how to numerically construct this projected flow in a stable and robust manner in particular the method avoids representation singularities by in practice choosing the best coordinate patch representation for the flow as it evolves the method is analytic in the spectral parameter and of complexity bounded by the order of the spectral problem cubed for large systems it represents a competitive method to those recently developed that are based on continuous orthogonalization we demonstrate this by comparing the two methods in three applications boussinesq solitary waves autocatalytic travelling waves and the ekman boundary layer

sectional rank and cohomology abstract donovan s conjecture implies a bound on the dimensions of cohomology groups in terms of the size of a sylow p subgroup and we give a proof of a stronger bound in terms of sectional p rank for dim h 1 g v we also prove a reduction theorem for higher cohomology

stabilization of daes and invariant manifolds many methods have been proposed for the stabilization of higher index differential algebraic equations daes such methods often involve constraint differentiation and problem stabilization thus obtaining a stabilized index reduction a popular method is baumgarte stabilization but the choice of parameters to make it robust is unclear in practice here we explain why the baumgarte method may run into trouble we then show how to improve it we further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature our approach is to i consider stabilization of odes with invariants ii discretize the stabilizing term in a simple way generally different from the ode discretization and iii use orthogonal projections whenever possible the best methods thus obtained are related to methods of coordinate projection we discuss them and make concrete algorithmic suggestions

decomposition matrices for low rank unitary groups we study the decomposition matrices of the unipotent blocks of finite special unitary groups sun q for unitary primes larger than n up to few unknown entries we give a complete solution for n 2 10 we also prove a general result for two column partitions when divides q 1 this is achieved using projective modules coming from the adic cohomology of deligne lusztig varieties

on the modular composition factors of the steinberg representation abstract let g be a finite group of lie type and st k be the steinberg representation of g defined over a field k we are interested in the case where k has prime characteristic l and st k is reducible tinberg has shown that the socle of st k is always simple we give a new proof of this result in terms of the hecke algebra of g with respect to a borel subgroup and show how to identify the simple socle of st k among the principal series representations of g furthermore we determine the composition length of st k when g gl n q or g is a finite classical group and l is a so called linear prime

long time stability of multi dimensional noncharacteristic viscous boundary layers we establish long time stability of multi dimensional noncharacteristic boundary layers of a class of hyperbolic parabolic systems including the compressible navier stokes equations with inflow outflow boundary conditions under the assumption of strong spectral or uniform evans stability evans stability has been verified for small amplitude layers by gues metivier williams and zumbrun for large amplitude layers it may be efficiently checked numerically as done in the one dimensional case by costanzino humpherys nguyen and zumbrun

on donovan s conjecture abstract this article contains reduction theorems for some weaker variants of donovan s conjecture which supposes that for every finite group d of prime order p there are only finitely many morita equivalence classes of p blocks of finite groups having defect groups isomorphic to d i when restricting the conjecture to the case of abelian defect groups it can be reduced to only considering p blocks of a class of groups that are close to being direct products of simple groups ii when restricting the conjecture to the case of principal blocks of finite groups having an abelian sylow p subgroup it can be reduced to only considering p blocks of simple groups a weaker version of donovan s conjecture would suppose that there are only finitely many cartan matrices of p blocks of finite groups with defect group isomorphic to a given one iii this conjecture can be reduced to only considering p blocks of quasisimple groups with centers having orders prime to p

numerical testing of the stability of viscous shock waves a new theoretical evans function condition is used as the basis of a numerical test of viscous shock wave stability accuracy of the method is demonstrated through comparison against exact solutions a convergence study and evaluation of approximate error equations robustness is demonstrated by applying the method to waves for which no current analytic results apply highly nonlinear waves from the cubic model and strong shocks from gas dynamics an interesting aspect of the analysis is the need to incorporate features from the analytic evans function theory for purposes of numerical stability for example we find it necessary for numerical accuracy to solve odes on the space of wedge products

insertion deletion correction with spectral nulls levenshtein 1966 proposed a class of single insertion deletion correcting codes based on the number theoretic construction due to varshamov and tenengolt s 1965 we present several interesting results on the binary structure of these codes and their relation to constrained codes with nulls in the power spectral density function one surprising result is that the higher order spectral null codes of immink and beenker 1987 are sub codes of balanced levenshtein codes other spectral null sub codes with similar coding rates may also be constructed we furthermore present some coding schemes and spectral shaping markers which alleviate the fundamental restriction on levenshtein s codes that the boundaries of each codeword should be known before insertion deletion correction can be effected

computing the maslov index of solitary waves part 2 phase space with dimension greater than four this paper extends the theory of the maslov index of solitary waves in part 1 to the case where the phase space is of dimension greater than four the starting point is hamiltonian pdes in one space dimension and time whose steady part is a hamiltonian ode with a phase space of dimension six or greater this steady hamiltonian ode is the main focus of the paper homoclinic orbits of the steady ode represent solitary waves of the pde and one of the properties of the homoclinic orbits is the maslov index we develop formulae for the maslov index the maslov angle and its subangles in an exterior algebra framework and develop numerical algorithms to compute them in addition a new numerical approach based on a discrete qr algorithm is proposed the maslov index is of interest for classifying solitary waves and as an indicator of stability or instability of the solitary wave in the time dependent problem the theory is applied to a class of reaction diffusion equations the longwave shortwave resonance equations and the seventh order kdv equation

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

spectral stability of ideal gas shock layers extending recent results in the isentropic case we use a combination of asymptotic ode estimates and numerical evans function computations to examine the spectral stability of shock wave solutions of the compressible navier stokes equations with ideal gas equation of state our main results are that in appropriately rescaled coordinates the evans function associated with the linearized operator about the wave i converges in the large amplitude limit to the evans function for a limiting shock profile of the same equations for which internal energy vanishes at one end state and ii has no unstable positive real part zeros outside a uniform ball thus the rescaled eigenvalue ode for the set of all shock waves augmented with the nonphysical limiting case form a compact family of boundary value problems that can be conveniently investigated numerically an extensive numerical evans function study yields one dimensional spectral stability independent of amplitude for gas constant in 1 2 3 and ratio of heat conduction to viscosity coefficient within 0 2 5 1 4 1 47 for air other values may be treated similarly but were not considered the method of analysis extends also to the multi dimensional case a direction that we shall pursue in a future work

the refined inviscid stability condition and cellular instability of viscous shock waves abstract combining the work of serre and zumbrun benzoni gavage serre and zumbrun and texier and zumbrun we propose as a mechanism for the onset of cellular instability of viscous shock and detonation waves in a finite cross section duct the violation of the refined planar stability condition of zumbrun serre a viscous correction of the inviscid planar stability condition of majda more precisely we show for a model problem involving flow in a rectangular duct with artificial periodic boundary conditions that transition to multidimensional instability through violation of the refined stability condition of planar viscous shock waves on the whole space generically implies for a duct of sufficiently large cross section a cascade of hopf bifurcations involving more and more complicated cellular instabilities the refined condition is numerically calculable as described by benzoni gavage serre zumbrun

insertion deletion detecting codes and the boundary problem insertion deletion detecting codes were introduced by konstantinidis in this paper we define insertion deletion detecting codes in a slightly different manner and based on this definition we introduce multiple deletion and multiple insertion detecting codes it is shown that these codes which are systematic are optimal in the sense that there exists no other systematic multiple deletion insertion detecting codes with a better rate one of the limitations of number theoretic code constructions intended to correct insertion deletion errors e g the levenshtein code is that they require received codeword boundaries to be known in order to successfully decode in literature a number of schemes have been proposed to deal with this problem we show how insertion deletion detecting codes as presented in this paper can be used to improve and or extend some of these schemes

systematic encoding of the varshamov tenengol ts codes and the constantin rao codes the maximum number of information bits that can be encoded systematically by the number theoretic codes of varshamov and tenengol ts 1965 is determined this number is also studied for the more general class of the group theoretic constantin rao 1979 codes although these codes are at least as large as the varshamov tenengol ts codes it is shown that the number of bits that can be systematically encoded using a constantin rao code does not exceed the number of bits that can be systematically encoded using a varshamov tenengol ts code in fact in many cases the largest constantin rao code has the least number of bits that can be systematically encoded

the wentzel kramers brillouin approximation of semiclassical eigenvalues of the zakharov shabat problem the wkb approximation of semiclassical eigenvalues of the non self adjoint zakharov shabat problem is a standard element in the theory of the integrable focusing nonlinear schrodinger equation this approximation is based on a reformulation of the eigenvalue problem as the eigenvalue problem for a self adjoint schrodinger operator with a correction term that depends on the spectral parameter the approximation results from neglecting the correction term we perform a numerical experiment which gives new evidence to support the validity of this procedure in the context of the semiclassical limit problem for the focusing nonlinear schrodinger equation in particular our results suggest that the rate of convergence of the approximate eigenvalues to the true ones is of the order of the square of the small parameter this information is relevant to the task of rigorously incorporating this approximation into the asymptotic analysis of the singular limit for the focusing nonlinear schrodinger equation

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

the weighted fusion category algebra and the q schur algebra for gl 2 q abstract we show that the weighted fusion category algebra of the principal 2 block b 0 of gl 2 q is the quotient of the q schur algebra s 2 q by its socle for q an odd prime power as a consequence we get a canonical bijection between the set of isomorphism classes of simple k gl 2 q b 0 modules and the set of conjugacy classes of b 0 weights in this case

stability of viscous detonations for majda s model abstract using analytical and numerical evans function techniques we examine the spectral stability of strong detonation wave solutions of a version of majda s scalar model for a reacting gas mixture with an arrhenius type ignition function we introduce an energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane and we use a numerical approximation of the evans function to search for possible unstable eigenvalues in this region our results show for the parameter values tested that these waves are spectrally stable combining these numerical results with the pointwise green function analysis of lyng raoofi texier zumbrun g lyng m raoofi b texier k zumbrun pointwise green function bounds and stability of combustion waves j differential equations 233 2 2007 654 698 we conclude that these waves are nonlinearly stable this represents the first demonstration of nonlinear stability for detonation wave solutions of a majda type model without a smallness assumption notably our results indicate that for this simplified scalar model there does not occur either in a normal parameter range or in the limit of high activation energy hopf bifurcation to galloping or pulsating solutions as is observed in the full reactive navier stokes equations this answers in the negative for this model a question posed by majda as to whether such scalar detonation models capture this aspect of detonation behavior

numerical computation of solitary waves in infinite cylindrical domains the numerical computation of solitary waves to semilinear elliptic equations in infinite cylindrical domains is investigated rather than solving on the infinite cylinder the equation is approximated by a boundary value problem on a finite cylinder convergence and stability results for this approach are given it is also shown that galerkin approximations can be used to compute solitary waves of the elliptic problem on the finite cylinder in addition it is demonstrated that the aforementioned procedures simplify in cases where the elliptic equation admits an additional reversibility structure finally the theoretical predictions are compared with numerical computations in particular post buckling of an infinitely long cylindrical shell under axial compression is considered it is shown numerically that for a fixed spatial truncation the error in the truncation scales with the length of the cylinder as predicted theoretically

computing lyapunov exponents on a stiefel manifold the problem of numerical computation of a few lyapunov exponents les of finite dimensional dynamical systems is considered from the viewpoint of the differential geometry of stiefel manifolds whether one computes one many or all les of a continuous dynamical system by time integration discrete or continuous orthonormalization is essential for stable numerical integration a differential geometric view of continuous orthogonalization suggests that one restricts the linearized vectorfield to a stiefel manifold however the stiefel manifold is not in general an attracting submanifold of the ambient euclidean space it is a constraint manifold with a weak numerical invariant new numerical algorithms for this problem are then designed which use the fiber bundle characterization of these manifolds this framework leads to a new class of systems for continuous orthogonalization which have strong numerical invariance properties and the strong skew symmetry property numerical integration of these new systems with geometric integrators leads to a new class of numerical methods for computing a few les which preserve orthonormality to machine accuracy this idea is also taken a step further by making the stiefel manifold an attracting invariant manifold in which case standard explicit runge kutta algorithms can be used this leads to an algorithm which requires only marginally more computation than a standard explicit integration without orthogonalization these class of methods should be particularly effective for computing a few les for large dimension dynamical systems the new schemes are straightforward to implement a test case is presented for illustration and an example from dynamical systems is presented where a few les are computed for an array of coupled oscillators

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

a variable length approach to moment balancing moment balancing templates have been proposed for channels with a small probability of an insertion deletion several orders smaller than additive errors that add a minimal amount of redundancy these templates are essentially a systematic way of encoding number theoretic codes primarily levenshtein s s 1 insertion deletion code moment balancing templates proposed up to this point have been of fixed length in this paper it is shown that by using variable length templates it is possible to obtain better performance than the optimal fixed length moment balancing template here performance is defined as the amount of redundancy which includes the moment balancing bits and the marker that needs to be added

stability of isentropic navier stokes shocks in the high mach number limit by a combination of asymptotic ode estimates and numerical evans function calculations we establish stability of viscous shock solutions of the isentropic compressible navier stokes equations with law pressure i in the limit as mach number m goes to infinity for any 1 proved analytically and ii for m 2 500 gamma in 1 2 5 or m 13 000 gamma in 2 5 3 demonstrated numerically this builds on and completes earlier studies by matsumura nishihara and barker humpherys rudd zumbrun establishing stability for low and intermediate mach numbers respectively indicating unconditional stability independent of shock amplitude of viscous shock waves for law gas dynamics in the range gamma in 1 3 other values may be treated similarly but have not been checked numerically the main idea is to establish convergence of the evans function in the high mach number limit to that of a pressureless or infinitely compressible gas with additional upstream boundary condition determined by a boundary layer analysis recall that low mach number behavior is formally incompressible

modular irreducibility of cuspidal unipotent characters we prove a long standing conjecture of geck which predicts that cuspidal unipotent characters remain irreducible after ell reduction to this end we construct a progenerator for the category of representations of a finite reductive group coming from generalised gelfand graev representations this is achieved by showing that cuspidal representations appear in the head of generalised gelfand graev representations attached to cuspidal unipotent classes as defined and studied in geck and malle j lond math soc 2 53 63 78 1996

nonlinear modulational stability of periodic traveling wave solutions of the generalized kuramoto sivashinsky equation abstract in this paper we consider the spectral and nonlinear stabilities of periodic traveling wave solutions of a generalized kuramoto sivashinsky equation in particular we resolve the long standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized integrable perturbations our analysis is based upon detailed estimates of the linearized solution operator which are complicated by the fact that the necessarily essential spectrum of the associated linearization intersects the imaginary axis at the origin we carry out a numerical evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves in close agreement with previous numerical studies of frisch she thual bar nepomnyashchy chang demekhin kopelevich and others carried out by other techniques we also compare predictions of the associated whitham modulation equations which formally describe the dynamics of weak large scale perturbations of a periodic wave train with numerical time evolution studies demonstrating their effectiveness at a practical level for the reader s convenience we include in an appendix the corresponding treatment of the swift hohenberg equation a nonconservative counterpart of the generalized kuramoto sivashinsky equation for which the nonlinear stability analysis is considerably simpler together with numerical evans function analyses extending spectral stability analyses of mielke and schneider

the instability of the hocking stewartson pulse and its geometric phase in the hopf bundle this work demonstrates an innovative numerical method for counting and locating eigenvalues with the evans function utilizing the geometric phase in the hopf bundle the technique calculates the winding of the evans function about a contour in the spectral plane describing the eigenvalues enclosed by the contour for the hocking stewartson pulse of the complex ginzburg landau equation locating eigenvalues with the geometric phase in the hopf bundle was proposed by way 2009 and proven by grudzien et al 2016 way demonstrated his proposed method for the hocking stewartson pulse and this manuscript redevelops this example as in the proof of the method in grudzien et al 2016 modifying his numerical shooting argument and introduces new numerical results concerning the phase transition

nonclassical multidimensional viscous and inviscid shocks extending our earlier work on lax type shocks of systems of conservation laws we establish existence and stability of curved multidimensional shock fronts in the vanishing viscosity limit for general lax or undercompressive type shock waves of nonconservative hyperbolic systems with parabolic regularization the hyperbolic equations may be of variable multiplicity and the parabolic regularization may be of real or partially parabolic type we prove an existence result for inviscid nonconservative shocks that extends to multidimensional shocks a one dimensional result of x lin proved by quite different methods in addition we construct families of smooth viscous shocks converging to a given inviscid shock as viscosity goes to zero thereby justifying the small viscosity limit for multidimensional nonconservative shocks in our previous work on shocks we made use of conservative form especially in parts of the low frequency analysis thus most of the new analysis of this paper is concentrated in this area by adopting the more general nonconservative viewpoint we are able to shed new light on both the viscous and inviscid theory for example we can now provide a clearer geometric motivation for the low frequency analysis in the viscous case also we show that one may in the treatment of inviscid stability of nonclassical and or nonconservative shocks remove an apparently restrictive technical assumption made by mokrane and coulombel in their work on respectively shocktype nonconservative boundary problems and conservative undercompressive shocks another advantage of the nonconservative perspective is that lax and undercompressive shocks can be treated by exactly the same analysis

continuation of invariant subspaces in large bifurcation problems we summarize an algorithm for computing a smooth orthonormal basis for an invariant subspace of a parameter dependent matrix and describe how to extend it for numerical bifurcation analysis we adapt the continued subspace to track behavior relevant to bifurcations and use projection methods to deal with large problems to test our ideas we have integrated our code into matcont a program for numerical continuation and bifurcation analysis

the orr sommerfeld equation on a manifold the most effective and widely used methods for integrating the orr sommerfeld equation by shooting are the continuous orthogonalization method and the compound matrix method in this paper we consider this problem from a differential geometric point of view a new definition of orthogonalization is presented restriction of the orr sommerfeld to a complex stiefel manifold and this definition leads to a new formulation of continuous orthogonalization which differs in a precise and interesting geometric way from existing orthogonalization routines present orthogonalization methods based on davey9s algorithm are shown to have a different differential geometric interpretation restriction of the orr sommerfeld equation to a complex grassmanian manifold this leads us to introduce the concept of a grassmanian integrator which preserves linear independence and not necessarily orthogonality using properties of grassmanian manifolds and their tangent spaces a new grassmanian integrator is introduced that generalizes davey9s algorithm furthermore it is shown that the compound matrix method is a dual grassmanian integrator it uses plucker coordinates for integrating on a grassmanian manifold and this characterization suggests a new algorithm for constructing the compound matrices extension of the differential geometric framework to general systems of linear ordinary differential equations is also discussed

on the shock wave spectrum for isentropic gas dynamics with capillarity abstract we consider the stability problem for shock layers in slemrod s model of an isentropic gas with capillarity we show that these traveling waves are monotone in the weak capillarity case and become highly oscillatory as the capillarity strength increases using a spectral energy estimate we prove that small amplitude monotone shocks are spectrally stable we also show that monotone shocks have no unstable real spectrum regardless of amplitude this implies that any instabilities of these monotone traveling waves if they exist must occur through a hopf like bifurcation where one or more conjugate pairs of eigenvalues cross the imaginary axis we then conduct a systematic numerical evans function study which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths in particular we show that the transition from monotone to nonmonotone profiles does not appear to trigger any instabilities

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

galloping instability of viscous shock waves abstract motivated by physical and numerical observations of time oscillatory galloping spinning and cellular instabilities of detonation waves we study poincare hopf bifurcation of traveling wave solutions of viscous conservation laws the main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum preventing standard reduction to a finite dimensional center manifold we overcome this by direct lyapunov schmidt reduction using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap the key computation is a space time stability estimate on the transverse linearized solution operator reminiscent of duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

one dimensional stability of parallel shock layers in isentropic magnetohydrodynamics abstract extending investigations of barker humpherys lafitte rudd and zumbrun for compressible gas dynamics and freistuhler and trakhinin for compressible magnetohydrodynamics we study by a combination of asymptotic ode estimates and numerical evans function computations the one dimensional stability of parallel isentropic magnetohydrodynamic shock layers over the full range of physical parameters shock amplitude strength of imposed magnetic field viscosity magnetic permeability and electrical resistivity for a law gas with 1 3 other values may be treated similarly but were not checked numerically depending on magnetic field strength these shocks may be of fast lax intermediate overcompressive or slow lax type however the shock layer is independent of magnetic field consisting of a purely gas dynamical profile in each case our results indicate stability interesting features of the analysis are the need to renormalize the evans function in order to pass continuously across parameter values where the shock changes type or toward the large amplitude limit at frequency 0 and the systematic use of winding number computations on riemann surfaces

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

evans function and fredholm determinants we explore the relationship between the evans function transmission coefficient and fredholm determinant for systems of first order linear differential operators on the real line the applications we have in mind include linear stability problems associated with travelling wave solutions to nonlinear partial differential equations for example reaction diffusion or solitary wave equations the evans function and transmission coefficient which are both finite determinants are natural tools for both analytic and numerical determination of eigenvalues of such linear operators however inverting the eigenvalue problem by the free state operator generates a natural linear integral eigenvalue problem whose solvability is determined through the corresponding infinite fredholm determinant the relationship between all three determinants has received a lot of recent attention we focus on the case when the underlying fredholm operator is a trace class perturbation of the identity our new results include i clarification of the sense in which the evans function and transmission coefficient are equivalent and ii proof of the equivalence of the transmission coefficient and fredholm determinant in particular in the case of distinct far fields

numerical implementation of complex orthogonalization parallel transport on stiefel bundles and analyticity abstract numerical integration of complex linear systems of odes depending analytically on an eigenvalue parameter are considered complex orthogonalization which is required to stabilize the numerical integration results in non analytic systems it is shown that properties of eigenvalues are still efficiently recoverable by extracting information from a non analytic characteristic function the orthonormal systems are constructed using the geometry of stiefel bundles different forms of continuous orthogonalization in the literature are shown to correspond to different choices of connection one form on the stiefel bundle for the numerical integration gauss legendre runge kutta algorithms are the principal choice for preserving orthogonality and performance results are shown for a range of glrk methods the theory and methods are tested by application to example boundary value problems including the orr sommerfeld equation in hydrodynamic stability

pointwise green function bounds and stability of combustion waves abstract generalizing similar results for viscous shock and relaxation waves we establish sharp pointwise green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both majda s model and the full reacting compressible navier stokes equations with artificial viscosity with general multi species reaction and reaction dependent equation of state under the necessary conditions of strong spectral stability i e stable point spectrum of the linearized operator about the wave transversality of the profile as a connection in the traveling wave ode and hyperbolic stability of the associated chapman jouguet square wave approximation notably our results apply to combustion waves of any type weak or strong detonations or deflagrations reducing the study of stability to verification of a readily numerically checkable evans function condition together with spectral results of lyng and zumbrun this gives immediately stability of small amplitude strong detonations in the small heat release i e fluid dynamical limit simplifying and greatly extending previous results obtained by energy methods by liu ying and tesei tan for majda s model and the reactive navier stokes equations respectively

numerical exterior algebra and the compound matrix method the compound matrix method which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k 2 3 dimensional subspaces of mathbb c n by using compound matrices as coordinates is reformulated in a coordinate free way using exterior algebra spaces bigwedge k mathbb c n

scopes reduction and morita equivalence classes of blocks in finite classical groups ii abstract we extend our work of j algebra 230 2000 378 423 on unipotent blocks of classical groups to blocks of orthogonal groups containing characters labelled by degenerate symbols we also show that for classical groups in the linear prime case there are only finitely many morita types of unipotent blocks with a given defect group