invariant manifolds of partial functional differential equations this paper is concerned with the existence smoothness and attractivity of invariant manifolds for evolutionary processes on general banach spaces when the nonlinear perturbation has a small global lipschitz constant and locally ck smooth near the trivial solution such a nonlinear perturbation arises in many applications through the usual cut off procedure but the requirement in the existing literature that the nonlinear perturbation is globally ck smooth and has a globally small lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut off function our general results are illustrated by and applied to partial functional differential equations for which the phase space c r 0 x where r 0 and x being a banach space has no smooth inner product structure and for which the validity of variation of constants formula is still an interesting open problem

on the structure of specht modules in the principal block of fσ3p abstract let f be a field of characteristic p at least 5 we study the loewy structures of specht modules in the principal block of f 3 p we show that a specht module in the block has loewy length at most 4 and composition length at most 14 furthermore we classify which specht modules have loewy length 1 2 3 or 4 produce a specht module having 14 composition factors describe the second radical layer and the socle of the reducible specht modules and prove that if a specht module corresponds to a partition that is p regular and p restricted then the head of the specht module does not extend the socle

stabilization of a boussinesq system of kdv kdv type a family of boussinesq systems has recently been proposed by bona chen and saut in j l bona m chen j c saut boussinesq equations and other systems for small amplitude long waves in nonlinear dispersive media i derivation and linear theory j nonlinear sci 12 4 2002 283 318 to describe the two way propagation of small amplitude gravity waves on the surface of water in a canal in this paper we investigate the boundary stabilization of the boussinesq system of kdv kdv type posed on a bounded domain we design a two parameter family of feedback laws for which the solutions issuing from small data are globally defined and exponentially decreasing in the energy space

a non homogeneous boundary value problem for the korteweg de vries equation posed on a finite domain ii abstract studied here is an initial and boundary value problem for the korteweg de vries equation posed on a bounded interval with nonhomogeneous boundary conditions this particular problem arises naturally in certain circumstances when the equation is used as a model for waves and a numerical scheme is needed it is shown here that this initial boundary value problem is globally well posed in the l 2 based sobolev space h s 0 1 for any s 0 in addition the mapping that associates to appropriate initial and boundary data the corresponding solution is shown to be analytic as a function between appropriate banach spaces

sinc galerkin method for solving biharmonic problems there are many techniques available to numerically solve the biharmonic equation in this paper we show that the sinc galerkin method is a very effective tool in numerically solving this equation hermite interpolation is used to treat the nonhomogeneous boundary conditions our method is tested on examples and comparisons with other methods are made it is shown that the sinc galerkin method yields good results even when singularities occur at the boundaries

convergence analysis of the chebyshev legendre spectral method for a class of fredholm fractional integro differential equations abstract in this paper we propose and analyze a spectral chebyshev legendre approximation for fractional order integro differential equations of fredholm type the fractional derivative is described in the caputo sense our proposed method is illustrated by considering some examples whose exact solutions are available we prove that the error of the approximate solution decays exponentially in l 2 norm

3 2 pairs of symmetric group algebras and their intermediate defect 4 blocks abstract in this paper we study 3 2 pairs of symmetric group algebras and their intermediate block in detail the aim is to understand how one block of a 3 2 pair can inherit two properties the ext quiver being bipartite and the principal indecomposable modules having a common loewy length 7 from the other we establish some sufficient conditions for this inheritance and verify these conditions for some special blocks

the ordinary quiver of a weight three block of the symmetric group is bipartite abstract suppose f is a field of characteristic p 5 and that b is a p block of the symmetric group s n of defect 3 we prove that the ext 1 quiver of b is bipartite with the bipartition being described in a simple way using the leg lengths of p hooks of partitions

efficient spectral galerkin methods for systems of coupled second order equations and their applications we construct in this paper two efficient spectral galerkin algorithms for solving systems of n coupled second order equations the computational complexity of these algorithms is essentially n times the cost of solving one second order equation we present numerical results which illustrate the accuracy and flexibility of these algorithms as well as several interesting and challenging applications including in particular a number of high order nonlinear parabolic type equations

hecke algebras of classical type and their representation type the purpose of this article is to determine the representation type for all of the hecke algebras of classical type to do this we combine methods from our previous work which is used to obtain information on their gabriel quivers and recent advances in the theory of finite dimensional algebras the principal computation is for hecke algebras of type b with two parameters then we show that the representation type of hecke algebras is governed by their poincare polynomials

a fast direct solver for the biharmonic problem in a rectangular grid we present a fast direct solver methodology for the dirichlet biharmonic problem in a rectangle the solver is applicable in the case of the second order stephenson scheme j w stephenson j comput phys 55 1984 pp 65 80 as well as in the case of a new fourth order scheme which is discussed in this paper and is based on the capacitance matrix method b l buzbee and f w dorr siam j numer anal 11 1974 pp 1136 1150 p bjorstad siam j numer anal 20 1983 pp 59 71 the discrete biharmonic operator is decomposed into two components the first is a diagonal operator in the eigenfunction basis of the laplacian to which the fft algorithm is applied the second is a low rank perturbation operator given by the capacitance matrix which is due to the deviation of the discrete operators from diagonal form the sherman morrison formula g h golub and c f van loan matrix computations 3rd ed john hopkins university press baltimore md 1996 is applied to obtain a fast solution of the resulting linear system of equations

asymptotic stability of a nonlinear korteweg de vries equation with critical lengths abstract we study an initial boundary value problem of a nonlinear korteweg de vries equation posed on the finite interval 0 2 k where k is a positive integer the whole system has dirichlet boundary condition at the left end point and both of dirichlet and neumann homogeneous boundary conditions at the right end point it is known that the origin is not asymptotically stable for the linearized system around the origin we prove that the origin is locally asymptotically stable for the nonlinear system if the integer k is such that the kernel of the linear korteweg de vries stationary equation is of dimension 1 this is for example the case if k 1

exact boundary controllability of the korteweg de vries equation we consider boundary control of the distributed parameter system described by the korteweg de vries kdv equation posed on a finite interval alpha leq x leq beta left begin array l u t u x uu x u xxx 0 u alpha t h 1 t qquad u beta t h 2 t qquad u x beta t h 3 t u end array right qquad for t geq 0 it is shown that by choosing appropriate control inputs h j t j 1 2 3 one can always guide the system from a given initial state phi in h s alpha beta to a given terminal state psi in h s alpha beta in the time period 0 t so long as phi and psi satisfy phi cdot w cdot 0 h s alpha beta leq delta quad mbox and quad psi cdot w cdot t h s alpha beta leq delta for some delta 0 independent of phi and psi where s geq 0 and w equiv w x t is a given smooth solution of the kdv equation this exact boundary controllability is established by considering a related initial value control problem of the kdv equation posed on the whole line r various recently discovered smoothing properties of the kdv equation have played important roles in our approach

output feedback stabilization of the linearized korteweg de vries equation with right endpoint controllers abstract in this paper we prove the output feedback stabilization for the linearized korteweg de vries kdv equation posed on a finite domain in the case the full state of the system cannot be measured we assume that there is a sensor at the left end point of the domain capable of measuring the first and second order boundary traces of the solution this allows us to design a suitable observer system whose states can be used for constructing boundary feedbacks acting at the right endpoint so that both the observer and the original plant become exponentially stable stabilization of the original system is proved in the l 2 sense while the convergence of the observer system to the original plant is also proved in higher order sobolev norms the standard backstepping approach used to construct a left endpoint controller fails and presents mathematical challenges when building right endpoint controllers due to the overdetermined nature of the related kernel models in order to deal with this difficulty we use the method of ozsari and batal 2019 which is based on using modified target systems involving extra trace terms in addition we show that the number of controllers and boundary measurements can be reduced to one with the cost of a slightly lower exponential rate of decay we provide numerical simulations illustrating the efficacy of our controllers

exact boundary controllability of the korteweg de vries equation around a non trivial stationary solution the exact boundary controllability of the non linear korteweg de vries equation on bounded domains is studied only the first spatial derivative at the right endpoint is assumed to be controlled in this case the exact controllability has been shown by rosier 1997 when the length l of the domain is not in the set n

exact controllability of a nonlinear korteweg de vries equation on a critical spatial domain we consider the boundary controllability problem for a nonlinear korteweg de vries equation with the dirichlet boundary condition we study this problem for a spatial domain with a critical length for which the linearized control system is not controllable in order to deal with the nonlinearity we use a power series expansion of second order we prove that the nonlinear term gives the local exact controllability around the origin provided that the time of control is large enough

symmetric group blocks of small defect has a specht module s and a p modular irreducible module d the decomposition number s d is de ned to be the composition multiplicity of d in s the following facts areknown about blocks of weight w a if w 0 or 1 then all the decomposition numbers of the block are 0 or 1 b if w 2 and p 2 then all the decomposition numbers of the block are 0 or 1 s2 c if w 4 then some decomposition numbers of a block can be greater than 1 even if p w moreover if w 0 1 or 2 then there is a known method for determining the decompositionnumbers r the situation for the case w 3 is still not properly understood in particular thedecomposition number s 2p 2 2p 2 p 1 1 d 3p 3 2p 1 is yet to be determined for p 5 this is just one of a collection of decomposition numbers forweight 3 which we are unable to evaluate our investigation of blocks of weight 3 grew out of an attempt to improve upon the earlier resultsof martin and russell mr by explicitly calculating the decomposition numbers this led us todiscover various errors and omissions in mr which place in doubt the claim made there that whenp w all the decomposition numbers are 0 or 1 note incidentally that if the decompositionnumbers for a given block are known to be 0 or 1 then the decomposition numbers can in principle be determined by applying schaper s theorem sch jm1 2 basic resultsby using a result of brundan and kleshchev we are able to improve upon the presentation ofseveral of the basic techniques used in s1 s2 mr for estimating decomposition numbers in orderto state these results recall that the diagram of a partition is the set of nodes i j 1 j

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

source algebras of blocks sources of simple modules and a conjecture of feit abstract we verify a finiteness conjecture of feit on sources of simple modules over group algebras for various classes of finite groups related to the symmetric groups

preconditioning a class of fourth order problems by operator splitting we develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space we consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two discrete linear second order elliptic operators and exploit this property in designing the preconditioners the underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available we propose symmetric and non symmetric preconditioners along with theory and numerical experiments they both document crucial properties of the preconditioners as well as their practical performance it is important to note that we neither need h s regularity s 1 of the continuous problem nor quasi uniform grids

rapid stabilization for a korteweg de vries equation from the left dirichlet boundary condition this paper deals with the stabilization problem for the korteweg de vries equation posed on a bounded interval the control acts on the left dirichlet boundary condition at the right end point dirichlet and neumann homogeneous boundary conditions are considered the proposed feedback law forces the exponential decay of the system under a smallness condition on the initial data moreover the decay rate can be tuned to be as large as desired the feedback control law is designed by using the backstepping method

martin s conjecture holds for weight 3 blocks of symmetric groups abstract we prove that martin s conjecture holds for weight 3 blocks of symmetric group algebras that if these blocks have abelian defect group then their projective indecomposable modules have a common radical length 7

filtrations in rouquier blocks of symmetric groups and schur algebras we study rouquier blocks of symmetric groups and schur algebras in detail and obtain explicit descriptions for the radical layers of the principal indecomposable weyl young and specht modules of these blocks at the same time the jantzen filtrations of the weyl modules are shown to coincide with their radical filtrations we also address the conjectures of martin lascoux leclerc thibon rouquier and james for these blocks

efficient spectral galerkin methods iii polar and cylindrical geometries we present in this paper several extremely efficient and accurate spectral galerkin methods for second and fourth order equations in polar and cylindrical geometries these methods are based on appropriate variational formulations which incorporate naturally the pole condition s in particular the computational complexities of the chebyshev galerkin method in a disk and the chebyshev legendre galerkin method in a disk or a cylinder are quasi optimal optimal up to a logarithmic term as an indication of efficiency the cpu time for the poisson solver on a disk by our chebyshev galerkin method is only about 70 of the corresponding finite difference code in fishpack

adjustment matrices for weight three blocks of iwahori hecke algebras abstract we compute the adjustment matrices for weight 3 blocks of iwahori hecke algebras of type a in characteristic 2 or 3 this enables all the decomposition numbers for weight 3 blocks to be calculated

legendre galerkin spectral element method for the biharmonic equations and its applications abstract a spectral element method based on the legendre galerkin approximation is presented to solve the two dimensional biharmonic equations rigorous error analysis is carried out to establish the convergence of the method by constructing appropriate basis functions which satisfy the boundary conditions of the differential equations the discrete variational formulation is reduced to linear system with sparse and symmetric matrices which can be efficiently solved by a fast schur complement method accuracy test is provided to confirm the convergence rate of the theoretical results finally the proposed method is applied to calculate the displacement of an elastic plate under a uniform applied load and stream function of zero reynolds number flow in a driven cavity the results compare well with established results

averaging principle for the higher order nonlinear schrodinger equation with a random fast oscillation this work concerns the problem associated with averaging principle for a higher order nonlinear schrodinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction diffusion equation evolving with respect to the fast time this model can be translated into a multiscale stochastic partial differential equations stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time scales to be more precise under suitable conditions we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear schrodinger equation is an average with respect to the stationary measure of the fast varying process finally by using the khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation and as a consequence the system can be reduced to a single higher order nonlinear schrodinger equation with a modified coefficient

the korteweg de vries equation on a periodic domain with singular point dissipation this paper considers the korteweg de vries kdv equation eqalignno u t uu x u xxx 0 0 0 quad u 0 x u 0 x and the periodic boundary conditions u t 1 u t 0 u xx t 0 u xx t 1 with an l 2 stabilizing control input implemented by a feedback mechanism u x t 1 alpha u x t 0 and alpha 1 it can be shown that the solutions conserve the volume u int 1 0u t x dx and the constant state u 0 possesses the smallest energy among solutions with same volume it has been proved that the solution of the system exists and approaches u 0 as t rightarrow infty when alpha not 1 2 this paper studies the case for alpha 1 2 and gives a proof of the existence and exponential decay of the solutions by deriving estimates of the corresponding green s function and using semigroup theory the method used here also works for the other cases with alpha 1

two approaches for the stabilization of nonlinear kdv equation with boundary time delay feedback this paper concerns the nonlinear korteweg de vries equation with boundary time delay feedback under appropriate assumption on the coefficients of the feedbacks delayed or not we first prove that this nonlinear infinite dimensional system is well posed for small initial data the main results of our study are two theorems stating the exponential stability of the nonlinear time delay system two different methods are employed a lyapunov function approach allowing to have an estimation on the decay rate but with a restrictive assumption on the length of the spatial domain of the kdv equation and an observability inequality approach with a contradiction argument for any noncritical lengths but without estimation on the decay rate some numerical simulations are given to illustrate the results

timely communication efficient algorithms for solving a fourth order equation with the spectral galerkin method we show that one can derive an o n 3 spectral galerkin method for fourth order biharmonic type elliptic equations based on the use of chebyshev polynomials the use of chebyshev polynomials provides a fast transform between physical and spectral space which is advantageous when a sequence of problems must be solved e g as part of a nonlinear iteration this improves the result of shen siam j sci comput 16 1995 pp 74 87 which reported an o n 4 algorithm inferior to the o n 3 method developed earlier shen siam j sci comput 15 1994 pp 1440 1451 based on legendre polynomials but less practical in the case of multiple problems we further compare our method with an improved implementation of the legendre galerkin method based on the same approach

the derived categories of some blocks of symmetric groups and a conjecture of broue abstract a well known conjecture of broue in the representation theory of finite groups involves equivalences of derived categories of blocks the aim of this paper is to verify this conjecture for defect 2 blocks of symmetric groups actually we prove for these blocks a refinement of broue s conjecture due to rickard

small time local stabilization for a korteweg de vries equation abstract this paper focuses on the local small time stabilization of a korteweg de vries equation on bounded interval thanks to a time varying dirichlet feedback law on the left boundary recently backstepping approach has been successfully used to prove the null controllability of the corresponding linearized system instead of carleman inequalities we use the adding an integrator technique to gain regularity on boundary control term which clears the difficulty from getting stabilization in small time

defect 3 blocks of symmetric group algebras abstract in this paper we study if the following properties are present in defect 3 blocks of symmetric group algebras k g n where k has characteristic p 5 the ext quiver is bipartite the p i m s have a common loewy length 7 we obtain a criteria for a defect 3 block b of k g n to inherit these properties from a defect 3 block b of k g n 1 with which it forms a 3 1 pair we also verify that the principal blocks of k g n 3 p n 4 p 1 have these properties

cartan matrices and morita equivalence for blocks of the symmetric groups theorem 1 the blocks of the symmetric groups with a given weight w 0 over a field of characteristic p can be collected into families according to the shapes of their cores each family consists of morita equivalent blocks the number of such families is at most flf 1 i l w 1 11 and each family contains a block of some g where n is less than or equal to p2 p 1 2 w 1 2 4 wp

stokes eigenmodes in square domain and the stream function vorticity correlation the stokes eigenmodes in the square are numerically determined and their symmetry properties are identified the spectra evolution laws are in excellent qualitative agreement with the theoretical asymptotic predictions proposed by constantin and foias in navier stokes equations university of chicago press 1988 k k o k the slopes are reported here and are found to be specific to the eigenmodes symmetry family the dynamic equilibria are analyzed and show a linear relationship between the vorticity and the stream function in the core of the eigenmodes these features of the stokes eigenmodes confined in the square are shared by the fully periodic stokes eigenmodes

exact controllability and stabilizability of the korteweg de vries equation in this paper we consider distributed control of the system described by the korteweg de vries equation i atu uaxu e3 u f on the interval 0 0 with periodic boundary conditions ii au 27r t ku o t k 0 1 2 where the distributed control f f x t is restricted so that the volume f u x t dx of the solution is conserved both exact controllability and stabilizibility questions are studied for the system in the case of open loop control if the control f is allowed to act on the whole spatial domain 0 27r it is shown that the system is globally exactly controllable i e for given t 0 and functions q x x with the same volume one can alway find a control f so that the system i ii has a solution u x t satisfying u x 0 0 x u x t x if the control f is allowed to act on only a small subset of the domain 0 27r then the same result still holds if the initial and terminal states b and 0 have small amplitude in a certain sense in the case of closed loop control the distributed control f is assumed to be generated by a linear feedback law conserving the volume while monotonically reducing f u x t 2dx the solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state as in the open loop control case a small amplitude assumption is needed if the control is allowed to act on only a small subdomain the smoothing property of the periodic linear kdv equation discovered recently by bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper

controllability of the korteweg de vries equation from the right dirichlet boundary condition in this paper we consider the controllability of the korteweg de vries equation in a bounded interval when the control operates via the right dirichlet boundary condition while the left dirichlet and the right neumann boundary conditions are kept to zero we prove that the linearized equation is controllable if and only if the length of the spatial domain does not belong to some countable critical set when the length is not critical we prove the local exact controllability of the nonlinear equation

an efficient direct parallel spectral element solver for separable elliptic problems an efficient direct parallel elliptic solver based on the spectral element discretization is developed the direct solver is based on a matrix decomposition approach which reduces multi dimensional separable problems to a sequence of one dimensional problems that can be efficiently handled by a static condensation process thanks to the spectral accuracy and the localized nature of a spectral element discretization this elliptic solver is spectrally accurate and can be efficiently parallelized and it can serve as an essential building block for large scale high performance solvers in computational fluid dynamics and computational materials science

general vertices in ordinary quivers for symmetric group algebras abstract in j pure appl algebra 142 1999 249 260 martin and russell construct part of the ordinary quiver of the principal block of s np in characteristic p they define the notion of a general vertex and show that around general vertices the quiver assumes an n dimensional lattice like structure here we use different methods to find more vertices in the quiver of this general type

decomposition numbers for weight three blocks of symmetric groups and iwahori hecke algebras let f be a field q a non zero element of f and h n h f q g n the iwahori hecke algebra of the symmetric group g n if b is a block of h n of e weight 3 and the characteristic of f is at least 5 we prove that the decomposition numbers for b are all at most 1 in particular the decomposition numbers for a p block of g n of defect 3 are all at most 1

averaging principle for stochastic korteweg de vries equation abstract averaging principle is a powerful tool for studying qualitative analysis of nonlinear dynamical systems in this paper we will establish an averaging principle for stochastic korteweg de vries equation under a general averaging condition with the help of this averaging principle we can establish an effective approximation for the solution of stochastic korteweg de vries equation this can tell us the asymptotic behavior of the solution and make the interaction between nonlinearity uncertainty and multiple scales more clear in order to obtain this averaging principle we need to establish the smoothing effect for the third order operator

a legendre spectral galerkin method for the biharmonic dirichlet problem a legendre spectral galerkin method is presented for the solution of the biharmonic dirichlet problem on a square the solution and its laplacian are approximated using the set of basis functions suggested by shen which are linear combinations of two legendre polynomials a schur complement approach is used to reduce the resulting linear system to one involving the approximation of the laplacian of the solution on the two vertical sides of the square the schur complement system is solved by a preconditioned conjugate gradient method or the cholesky method the total cost of the algorithm is o n3 numerical results demonstrate the spectral convergence of the method

equating decomposition numbers for different primes abstract this paper shows that certain decomposition numbers for the iwahori hecke algebras of the symmetric groups and the q schur algebras at different roots of unity in characteristic zero are equal to prove our results we first establish the corresponding theorem for the canonical basis of the level one fock space and then apply deep results of ariki and varagnolo vasserot