finite size scaling and universality in the spectrum of the quantum ising chain i periodic and antiperiodic boundary condition the spectrum of the quantum ising chain is studied in the finite size scaling limit for periodic and antiperiodic boundary conditions the finite size corrections are computed for all energy gaps the results of the model are used to test the privman fisher universality hypothesis and conformal theory at the critical point it is shown how to build the spectrum from the irreducible representations of the virasoro algebra in addition the author finds evidence for an unsuspected symmetry for systems both non critical and of finite size

equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

on the universality of crossing probabilities in two dimensional percolation six percolation models in two dimensions are studied percolation by sites and by bonds on square hexagonal and triangular lattices rectangles of widtha and heightb are superimposed on the lattices and four functions representing the probabilities of certain crossings from one interval to another on the sides are measured numerically as functions of the ratioa b in the limits set by the sample size and by the conventions and on the range of the ratioa b measured the four functions coincide for the six models we conclude that the values of the four functions can be used as coordinates of the renormalization group fixed point

linking first occurrence polynomials over f2 by steenrod operations abstract it is proved that for every irreducible representation l of the full matrix semigroup mn f 2 the first occurrence of l as a composition factor in the polynomial algebra p f 2 x1 xn is linked by a steenrod operation to the first occurrence of l as a submodule in p this steenrod operation is given explicitly as the image of an admissible monomial in the steenrod squares sqr under the canonical anti automorphism of the mod 2 steenrod algebra a the first occurrences of both kinds are also linked to higher degree occurrences of l by elements of the milnor basis of a

a boundary value technique for singular perturbation problems abstract a new way to solve singular perturbation problems is introduced it is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems easy to use modest problem preparation and ready computer implementation as with other methods the original problem is partitioned into inner and outer solution differential equation systems however asymptotic solution techniques are not employed the method is distinguished by the following facts the inner solution problem is solved as a two point boundary problem where the terminal boundary conditions are supplied by the solution of the outer solution problem in turn the outer solution differential equations are solved as an initial value system where the initial conditions are supplied by the inner solution at its terminal point the method is iterative on the terminal point of the inner solution problem three numerical examples are included

finite size corrections to matrix elements in a conformal theory applications to the magnetisation of the three state potts model using conformal invariance the authors calculate the leading corrections in 1 n to the magnetisation of the three state potts quantum chain the results are in good agreement with the numerical finite size results

nonabelian localization in equivariant k theory and riemann roch for quotients abstract we prove a localization formula in equivariant algebraic k theory for an arbitrary complex algebraic group acting with finite stabilizer on a smooth algebraic space this extends to nondiagonalizable groups the localization formulas of nielsen diagonalizably linearized coherent sheaves bull soc math france 102 1974 85 97 and thomason une formule de lefschetz en k theorie equivariante algebrique duke math j 68 3 1992 447 462 as an application we give a riemann roch formula for quotients of smooth algebraic spaces by proper group actions this formula extends previous work of toen theoremes de riemann roch pour les champs de deligne mumford k theory 18 1 1999 33 76 and the authors riemann roch for quotients and todd classes of simplicial toric varieties comm algebra 31 2003 3735 3752

the composition factors of fp x1 x2 x3 as a gl 3 p module let gl n p denote the general linear group over the prime field fp and let sd denote the right gl n p module consisting of all homogeneous polynomials of degree d in n variables x1 x2 x over f we wish to determine the simple composition factors of sd unfortunately satisfactory models for all the simple gl n p modules themselves are not known there are p p 1 of these simple modules up to isomorphism and all of them are known to occur in the symmetric algebra s cf sd the general problem is therefore a very difficult one in our present state of knowledge the purpose of this paper is to offer a solution to this problem in the case n 3 this extends the work of david carlisle who calculated in his manchester thesis 2 the occurrences of all the l dimensional modules and developed a method for dealing with the general case let us begin with some general remarks about the problem the case n 1 is trivial as s d detd the dth power of the determinant representation the case n 2 is also easily disposed of for instance by glover 9 the simple modules are the tensor products of the sd for d 1 and d p the module s is not simple for example the pth power sf of the natural module is a proper submodule of 411 0021 8693192 3 00

universal functions in finite size scaling a numerical study in the ising and potts q 3 universality class the universality of the correlation length order parameter and susceptibility amplitudes is studied in the vicinity of the critical point of two dimensional models on strips with periodic boundary conditions for the ising and potts q 3 universality class the results confirm privman fisher conjecture 1984

on the ext groups between weyl modules for gln abstract this paper studies extension groups between certain weyl modules for the algebraic group gl n over the integers main results include 1 a complete determination of ext groups between weyl modules whose highest weights differ by a single root and 2 determination of ext 1 between an exterior power of the defining representation and any weyl module the significance of these results for modular representation theory of gl n is discussed in several remarks notably the first result leads to a calculation of ext groups between neighboring weyl modules for gl n and also recovers the gl n case of a recent result of andersen some generalities about ext groups between weyl modules and a brief overview of known results about these groups are also included

universal flattening of frobenius for a variety x of positive characteristic and a nonnegative integer e we define its e th f blowup to be the universal flattening of the e iterated frobenius of x thus we have the sequence a set labeled by nonnegative integers of blowups of x under some condition the sequence stabilizes and leads to a nice for instance minimal or crepant resolution for tame quotient singularities the sequence leads to the g hilbert scheme 1 introduction in this paper we introduce and study a characteristic p variant of the higher nash blowup introduced in 26 let x be a d dimensional variety over a perfect field k of characteristic p 0 xsm its smooth locus and f e xe x the e iterated k linear frobenius for a point x xsm k t he fiber f e 1 x xe k is a 0 dimensional subscheme of length q d with q p e and identified with a k point of the hilbert scheme hilb qd xe of length q d sub schemes of xe definition 1 1 we define the e th f blowup f b e x to be the closure of the set f e 1 x x xsm

computational methods for reaction diffusion problems for fourth order ordinary differential equations with a small parameter at the highest derivative in this paper basically asymptotic numerical methods for solving singularly perturbed two point boundary value problems for fourth order ordinary differential equations of the form ey i v x b x y x c x y x f x x d 0 1 y 0 p y 0 q y 0 r y 1 s is considered here a prime denotes a differentiation with respect to x b x c x and f x are smooth functions b x b 0 0 c x c c 0 and 0 e 1 the above boundary value problem is transformed into an equivalent weakly coupled system of two first order ordinary differential equations subject to suitable initial conditions and one second order singularly perturbed ordinary differential equations subject to suitable boundary conditions in order to solve this system three computational methods are suggested in this paper in these methods first we find a zero order asymptotic approximation of the solution of the weakly coupled system then the system is decoupled by replacing the first component of the solution by its zero order asymptotic approximation of the solution in the second order equation then the second order equation is solved separately by three methods namely fitted operator method fitted mesh method and boundary value technique error estimates are derived and examples are provided to illustrate the methods

graded mesh difference schemes for singularly perturbed two point boundary value problems etude de l approximation numerique des operateurs differentiels sans points de retournement par des schemas aux differences finies compacts analyse de la stabilite

invariant subspaces of the ring of functions on a vector space over a finite field abstract iffqis the finite field of characteristicpand orderq ps let f q be the category whose objects are functors from finite dimensionalfq vector spaces tofq vector spaces and with morphisms the natural transformations between such functors a fundamental object in f q is the injectiveifqdefined by i f q v f q v s v x q x we determine the lattice of subobjects ofifq it is the distributive lattice associated to a certain combinatorially defined poset i p s whoseqconnected components are all infinite with one trivial exception an analysis of i p s reveals that every proper subobject of an indecomposable summand ofifqis finite thusifqis artinian filteringifqand i p s in various ways yields various finite posets and we recover the main results of papers by doty kovacs and krop on the structure ofs v xq overfq ands v over f p

approximate method for the numerical solution of singular perturbation problems we present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval it is motivated by the asymptotic behavior of singular perturbation problems the original problem is divided into inner and outer region problems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem in turn the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm the proposed method is iterative on the terminal point some numerical experiments have been included to demonstrate its applicability

a numerical method for boundary value problems for singularly perturbed fourth order ordinary differential equations singularly perturbed two point boundary value problems bvps for fourth order ordinary differential equations odes with a small positive parameter multiplying the highest derivative are considered a numerical method is suggested in this paper to solve such problems in this method the given bvp is transformed into a system of two odes subject to suitable boundary conditions then the domain of definition of the differential equation a closed interval is divided into three non overlapping sub intervals which we call them inner regions boundary layers and outer region then the de is solved in these intervals separately the solutions obtained in these regions are combined to give a solution in the entire interval to obtain terminal boundary conditions boundary values inside this interval we use mostly zero order asymptotic expansion of the solution of the bvp first linear equations are considered and then non linear equations to solve non linear equations newton s method of quasi linearization is applied the present method is demonstrated by providing examples the method is easy to implement and suitable for parallel computing

subalgebras of the steenrod algebra and the action of matrices on truncated polynomial algebras let a t be the subalgebra of the steenrod algebra generated by sqi i 2t we determine which a module indecomposable summands of f2 x1 xn are free over a t by first showing this to be equivalent to the group theoretic question of determining which irreducible modular representations of mn f2 or gln f2 occur as composition factors in the truncated polynomial algebra f2 x1 xn x12 xn2 the second question is then answered

cohomology and generic cohomology of specht modules for the symmetric group cohomology of specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the borel subgroup b of the general linear group gld k but this has never been exploited to prove new symmetric group results using work of doty on the submodule structure of symmetric powers of the natural gld k module together with work of andersen on cohomology for b and its frobenius kernels we prove new results about hi d s we recover work of james in the case i 0 then we prove two stability theorems one of which is a generic cohomology result for specht modules equating cohomology of sp with sp2 this is the first theorem we know relating specht modules s and sp the second result equates cohomology of s with s pa for large a

the module structure of a group action on a polynomial ring a finiteness theorem consider a group acting on a polynomial ring over a finite field we study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries for example any indecomposable module that appears as a summand must also appear in low degree and so the number of isomorphism types of such summands is finite there are also applications to invariant theory giving a priori bounds on the degrees of the generators

metabelian lie powers of the natural module for a general linear group abstract consider a free metabelian lie algebra m of finite rank r over an infinite field k of prime characteristic p given a free generating set m acquires a grading its group of graded automorphisms is the general linear group g l r k so each homogeneous component m d is a finite dimensional g l r k module the homogeneous component m 1 of degree 1 is the natural module and the other m d are the metabelian lie powers of the title this paper investigates the submodule structure of the m d in the main result a composition series is constructed in each m d and the isomorphism types of the composition factors are identified both in terms of highest weights and in terms of steinberg s twisted tensor product theorem their dimensions are also given it turns out that the composition factors are pairwise non isomorphic from which it follows that the submodule lattice is finite and distributive by the birkhoff representation theorem any such lattice is explicitly recognizable from the poset of its join irreducible elements the poset relevant for m d is then determined by exploiting a 1975 paper of yu a bakhturin on identical relations in metabelian lie algebras

critical probability and scaling functions of bond percolation on two dimensional random lattices we locate the critical probability of bond percolation on two dimensional random lattices as because of the symmetry with respect to permutation of the two axes for random lattices we expect that for an aspect ratio of unity and sufficiently large lattices the probability of horizontal spanning equals the probability of vertical spanning this is confirmed by our monte carlo simulations we show that the ideas of universal scaling functions and non universal metric factors can be extended to random lattices by studying the existence probability and the percolation probability p on finite square planar triangular and random lattices with periodic boundary conditions using a histogram monte carlo method our results also indicate that the metric factors may be the same between random lattices and planar triangular lattices provided that the aspect ratios are 1 and

a computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers singularly perturbed turning point problems tpps for second order ordinary differential equations des exhibiting twin boundary layers are considered in order to obtain numerical solution of these problems a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods in this method the given internal domain of definition of differential equation is divided into four subintervals and we solve the differential equation in each interval and combine the solution the implementation of the method on parallel architectures is mentioned error estimates are derived and numerical examples are given

fixed point functors for symmetric groups and schur algebras abstract let d be the symmetric group for 1 f m be the functor which takes a d module u to the space of fixed points u m which is naturally a module for d m this functor was previously used by the author to study cohomology of the symmetric group but little is known about it this paper initiates a study of f m first we relate it to james work on row and column removal and decomposition numbers for the schur algebra next we determine the image of dual specht modules permutation and twisted permutation modules and some young and twisted young modules under f m in particular f m acts as first row removal on dual specht modules s with 1 m and as first column removal on twisted young and twisted permutation modules corresponding to partitions with m parts finally we prove that determining f m on the young modules is equivalent to determining the decomposition numbers for the schur algebra

a computational method for solving boundary value problems for third order singularly perturbed ordinary differential equations singularly perturbed two point boundary value problems spbvps for third order ordinary differential equations odes with a small parameter multiplying the highest derivative are considered a numerical method is suggested in this paper to solve such problems in this method the given bvp is transformed into a system of two odes subject to suitable initial and boundary conditions then the domain of definition of the differential equation a closed interval is divided into two sub intervals which we call the inner region boundary layer and the outer region then the de is solved in these intervals separately the solutions obtained in these intervals are combined to give the solution in the whole interval to obtain boundary conditions at the transition points boundary values inside this interval we use mostly the zeroth order asymptotic expansion of the solution of the bvp or a suitable asymptotic expansion solution first the linear equations are considered and then the semi linear equations to solve semi linear equations newton s method of quasi linearisation is applied examples are provided to illustrate the method the method is easy to implement and suitable for parallel computing

modular representations of gl 3 fp symmetric squares and mod p cohomology of gl 3 z abstract we prove certain cases of nonabelian reciprocity between mod p galois representations and hecke eigenclasses in the mod p cohomology of gl 3 z using the symmetric square liftings from gl 2 and congruences on hecke eigenvalues we need to develop some modular representation theory for gl 3 z p in some cases we also need to invoke certain computer calculations to finish the proof

the module structure of a group action on a polynomial ring abstract for any representation of a p group g on a vector space of dimension 3 over a finite field k of characteristic p we show how the symmetric algebra regarded as a kg module can be expressed as a direct sum of kg modules each one of which is isomorphic to a summand in low degree it follows that for any group g only a finite number of isomorphism classes of summands can occur

the two dimensional tricritical ising model in an external magnetic field the two dimensional tricritical ising model with an order parameter perturbation is studied using two different lattice models the mass spectrum is found to be the same as in the 2d ising model with a magnetic field results for the scaling functions are in agreement with universality and for the lowest mass a relationship with the scaling function of the ising model is observed the numerical results do not support a recent proposal of dotsenko on the dynamical generation of extra terms in the perturbative calculation of scaling functions

weyl modules and the mod 2 steenrod algebra abstract this paper continues our study of the action of the mod 2 steenrod algebra a 2 on the polynomial algebra p n f 2 t 1 t n we obtain further partial results on the hit problem of f p peterson which asks for a minimal generating set for p n as an a 2 module we also study the structure of the quotient by the hit elements as a graded representation of the finite general linear group g n gl n f 2 i e as a module over the finite group algebra f 2 g n these results were obtained in previous work of the authors for the special case of the steinberg module for g n by extending the scalars to f 2 the algebraic closure of f 2 we obtain commuting actions of a 2 and g n on p n f 2 t 1 t n while this makes no essential difference to the representation theory of g n or to the hit problem it allows us to treat the action of g n on p n as the restriction of that of the algebraic group g n gl n f 2 in particular we make use of tilting modules for g n to show that for every irreducible representation l of g n a minimal set of a 2 generators of p n must contain a copy of the corresponding dual weyl module

asymptotic numerical methods for singularly perturbed fourth order ordinary differential equations of reaction diffusion type abstract singularly perturbed boundary value problems bvps for fourth order ordinary differential equations odes with a small positive parameter multiplying the highest derivative of the form ey iv x b x y x c x y x f x x d 0 1 y 0 p y 1 q y 0 r y 1 0 e 1 are considered the given fourth order bvp is transformed into a system of weakly coupled systems of two second order odes one without the parameter and the other with the parameter e multiplying the highest derivative and suitable boundary conditions in this paper computational methods for solving this system are presented in these methods we first find the zero order asymptotic approximation expansion of the solution of the weakly coupled system then the system is decoupled by replacing the first component of the solution by its zero order asymptotic approximation expansion of the solution in the second equation then the second equation is solved by the fitted operator method fitted mesh method and boundary value technique error estimates are derived and examples are provided to illustrate the methods

an analogue of serre s conjecture for galois representations and hecke eigenclasses in the mod p cohomology of gl n ℤ

a computational method for solving singular perturbation problems abstract a class of singularly perturbed two point boundary value problems for second order ordinary differential equations is considered in order to solve them a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods the proposed method is distinguished by the following facts first we divide the given interval the domain of definition of differential equation into two subintervals called inner and outer regions then we solve the differential equation over both the regions as two point boundary value problems the terminal boundary condition of the inner region is obtained using the zero order asymptotic expansion of the solution some numerical examples are given to illustrate the method

the morava k theories of some classifying spaces let p be a finite abelian p group with classifying space bp we compute in representation theoretic terms the morava k theories of the stable wedge summands of bp in particular we obtain a simple and purely group theoretic description of the rank of k s bg for any finite group g with an abelian p sylow subgroup a minimal amount of topology quickly reduces the problem to an algebraic one of analyzing truncated polynomial algebras as modular representations of the semigroup m 1 z p 1 main results for a fixed prime p there exists a sequence of cohomology theories k 1 k 2 with k s periodic with period 2 ps 1 these are the morava k theories 13 and generalize ordinary complex k theory in the sense that k 1 is one of the p 1 isomorphic summands of k z p recently it has been becoming clear that they play a central role in homotopy theory see for example the work of m hopkins and j smith 6 k s has further computational virtues k s x is always a free module over the coefficient ring k s and there is a kunneth isomorphism a major outstanding problem is to find good models for the spaces representing these theories related to this is the question of finding for finite groups g a group theoretic description for the rings k s bg analogous to atiyah s isomorphism 2 r g k bg the author together with j harris recently analyzed stable wedge decompositions of classifying spaces of abelian p groups 5 in this paper we compute in representation theoretic terms the morava k theories of the resulting wedge summands as a consequence we obtain a very simple and purely group theoretic description of the rank of k s bg for any finite group g with an abelian p sylow subgroup to state our results we establish some notation it is convenient to let k o be rational cohomology we let k s x always denote the reduced sth morava k theory of x if x is a space or spectrum we let k5 x equal the rank of k s x as a k s module and by abuse of notation we let ks g ks bg for a finite group g bg is the union of bg with a disjoint basepoint received bv the editors may 27 1986 1980 marhemat7es subject cl lssific ltion primary 55n22 secondary 20g05 research partially supported by the sloan foundation and the nsf t1987 american mathematical society 0002 9947 87 1 00 25 per page

numerical treatment of singularly perturbed two point boundary value problems we propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval this is a practical method and can be easily implemented on a computer the original problem is divided into inner and outer region differential equation systems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem tpbvp in turn the outer region problem is also solved as a tpbvp both these tpbvps are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions the stability of some recurrence relations involved in the algorithm is investigated the proposed method is iterative on the terminal point some numerical examples are included and the computational results are compared with exact solutions it is observed that the accuracy predicted can always be achieved with very little computational effort

structure theorems over polynomial rings given a polynomial ring r over a field k and a finite group g we consider a finitely generated graded rg module s we regard s as a kg module and show that various conditions on s are equivalent such as only containing finitely many isomorphism classes of indecomposable summands or satisfying a structure theorem in the sense of d karagueuzian p symonds the module structure of a group action on a polynomial ring a finiteness theorem preprint http www ma umist ac uk pas preprints

galois representations with conjectural connections to arithmetic cohomology in this paper we extend a conjecture of a ash and w sinnott relating niveau 1 galois representations to the mod p cohomology of congruence subgroups ofsln z to include galois representations of higher niveau we then present computational evidence for our conjecture in the case n 3 in the form of three dimensional galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture our examples include galois representations with nontrivial weight and level as well as irreducible three dimensional representations that are in no obvious way related to lower dimensional representations in addition we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture

on the cohomology of young modules for the symmetric group abstract the main result of this paper is an application of the topology of the space q x to obtain results for the cohomology of the symmetric group on d letters d with twisted coefficients in various choices of young modules and to show that these computations reduce to certain natural questions in representation theory the authors extend classical methods for analyzing the homology of certain spaces q x with mod p coefficients to describe the homology h d v d as a module for the general linear group gl v over an algebraically closed field k of characteristic p as a direct application these results provide a method of reducing the computation of ext d y y where y y are young modules to a representation theoretic problem involving the determination of tensor products and decomposition numbers in particular in characteristic two for many d a complete determination of h d y can be found this is the first nontrivial class of symmetric group modules where a complete description of the cohomology in all degrees can be given for arbitrary d the authors determine h i d y for i 0 1 2 an interesting phenomenon is uncovered namely a stability result reminiscent of generic cohomology for algebraic groups for each i the cohomology h i p a d y p a stabilizes as a increases the methods in this paper are also powerful enough to determine for any p and precisely when h d y 0 such modules with vanishing cohomology are of great interest in representation theory because their support varieties constitute the representation theoretic nucleus

universality of the excess number of clusters and the crossing probability function in three dimensional percolation extensive monte carlo simulations were performed to evaluate the excess number of clusters and the crossing probability function for three dimensional percolation on the simple cubic s c face centred cubic f c c and body centred cubic b c c lattices systems l l l0 with l0 l were studied for both bond s c f c c b c c and site f c c percolation the excess number of clusters q b per unit length was confirmed to be a universal quantity with a value q b 0 412 likewise the critical crossing probability in the l0 direction with periodic boundary conditions in thell plane was found to follow a universal exponential decay as a function of r d l 0 l for large r simulations were also carried out to find new precise values of the critical thresholds for site percolation on the f c c and b c c lattices yielding pc f c c d 0 199 2365 0 000 0010 pc b c c d 0 245 9615 0 000 0010 we also report the value pc s c d 0 311 6080 0 000 0004 for site percolation

higher algebraic k theory for actions of diagonalizable groups we study the k theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces our main result shows how to reconstruct the k theory ring of such an action from the k theory rings of the loci where the stabilizers have constant dimension we apply this to the calculation of the equivariant k theory of toric varieties and give conditions under which the merkurjev spectral sequence degenerates so that the equivariant k theory ring determines the ordinary k theory ring we also prove a very refined localization theorem for actions of this type

higher algebraic k theory of group actions with finite stabilizers we prove a decomposition theorem for the equivariantk theory of actions of ane group schemes g of nite type over a eld on regular separated noetherian algebraic spaces under the hypothesis that the actions have nite geometric stabilizers and satisfy a rationality condition together with a technical condition which holds e g for g abelian or smooth we reduce the problem to the case of a gln action and nally to a split torus action

a numerical method for singular perturbation problems arising in chemical reactor theory abstract a class of singularly perturbed two point boundary value problems for second order ordinary differential equations with mixed boundary conditions arising in chemical reactor theory is considered in order to solve them a numerical method is suggested in which an exponentially fitted difference scheme is combined with classical numerical methods the proposed method is distinguished by the following facts first we divide the given interval the domain of definition of the differential equation into two subintervals called outer and inner regions then we solve the differential equation over both the regions as two point boundary value problems the terminal boundary condition of the inner region is obtained using the zero order asymptotic expansion of the solution some numerical examples are given to illustrate the method

group actions on rings and the cech complex abstract we have previously shown that when a finite group g acts on a polynomial ring s in n variables over a finite field k only finitely many isomorphism classes of indecomposable k g modules occur as summands of s we have also shown that the regularity of the invariant subring s g is at most zero which has various consequences for example s g is generated in degrees at most n g 1 provided n g 2 both of these theorems depend on the structure theorem of karagueuzian and the author which is proved by means of a long and complicated calculation the aim of this paper is to prove these results using a more conceptual method

the asymptotic solution of a class of third order boundary value problems arising in the theory of thin film flows this paper studies boundary and interior layer phenomena exhibited by solutions of certain singularly perturbed third order boundary value problems which govern the motion of thin liquid films subject to viscous capillary and gravitational forces precise conditions specifying where and when the third order derivative terms in the differential equations can be neglected are derived and improved estimates for the actual solutions in terms of solutions of the lower order models are constructed the paper also contains a technique for replacing a third order problem with an asymptotically equivalent second order one that may have wider applicability

schur asymptotics of veronese syzygies we study the asymptotic behavior of veronese syzygies as representations of the general linear group for a fixed homological degree p of the syzygies we describe the exact asymptotic growth for the number of distinct irreducible representations and for the number of irreducible representations also counting multiplicities this shows that asymptotically veronese syzygies have a very rich algebraic and representation theoretic structure as the degree of the embedding grows

analytical and non analytical corrections to finite size scaling using conformal invariance the author calculates finite size corrections to the scaled spectra of one dimensional quantum chains at the critical point an explicit application to the ising and three state potts model gives good agreement with analytical and numerical results respectively

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

die gleichmaβige stabilitat singular gestorter diskreter und kontinuierlicher randwertprobleme in this paper we consider singular perturbation problems for ordinary differential operators of ordern and their discrete counterparts on arbitrary nonuniform grids we prove that the singularly perturbed initial value problem is stable uniformly in the perturbation parameter in both the continuous and the discrete case we use this result to characterize the stability of the corresponding continuous and discrete boundary value problems if the continuous problem is stable and if the consistency error is smaller than a certain constant the discrete problem is also stable

on the representations of the full matrix semigroup on homogeneous polynomials this paper is the successor of 11 in the closing theorem there we determined the composition factors of w for a regular i 1 6 11 here we are going to describe the structure of wa in the singular case the second theme of this paper is the criterion for distributivity of the m modules wcd since the necessary conditions for that established in 1 8 51 limit d to d p p 1 we just have to sort out the latter case this will be done in the third section of this paper for the convenience of the reader we recall some notation terms and definitions used in 11

further examples of the boundary value technique in singular perturbation problems illustration de la technique des valeurs frontieres dans les problemes a perturbations singulieres par des exemples varies

singular perturbations of boundary value problems for a class of third order nonlinear ordinary differential equations abstract this article shows the existence uniqueness and asymptotic estimates of solutions of the singularly perturbed boundary value problems for a class of the third order nonlinear ordinary differential equations with a small parameter e 0 ey x y y e satisfying the condition y 0 0 y 1 0 y 1 0 or y 0 0 y 1 0 y 1 0

asymptotic numerical methods for singularly perturbed fourth order ordinary differential equations of convection diffusion type singularly perturbed boundary value problems bvps for fourth order ordinary differential equations odes with a small positive parameter multiplying the highest derivative of the email protected i v x a x email protected x b x y x c x y x f x email protected d 0 1 y 0 p y 1 q y 0 r y 1 s are considered the given fourth order bvp is transformed into a system of weakly coupled system of two second order odes one without the parameter and the other with the parameter e multiplying the highest derivative and suitable boundary conditions in this paper computational methods for solving this system are presented in these methods we first find the zero order asymptotic approximation expansion of the solution of the weakly coupled system then the system is decoupled by replacing the first component of the solution by its zero order asymptotic approximation expansion of the solution in the second equation then the second equation is solved by the fitted operator method fom fitted mesh method fmm and boundary value technique bvt error estimates are derived and examples are provided to illustrate the methods

isomorphisms of finite invariant for enveloping algebras semi simple case letgbe a finite dimensional semi simple lie algebra u g its enveloping algebra andha finite subgroup ofaut u g letabe the invariant algebrauh in this article we prove that the lie algebragis given up to an isomorphism by the algebraa if we impose thathis a finite subgroup of the adjoint group ofgacting on the enveloping algebrau g then the algebraagives a unique group algebrac h ifg sl2 then the grouphcan be recovered froma

galois structure of homogeneous coordinate rings suppose g is a finite group acting on a projective scheme x over a commutative noetherian ring r we study the rg modules h 0 x f l n when n 0 and f and are coherent g sheaves on x such that l is an ample line bundle we show that the classes of these modules in the grothendieck group g 0 rg of all finitely generated rg modules lie in a finitely generated subgroup under various hypotheses we show that there is a finite set of indecomposable rg modules such that each h 0 x f l n is a direct sum of these indecomposables with multiplicities given by generalized hilbert polynomials for n 0

noncommutative resolution f blowups and d modules abstract we explain the isomorphism between the g hilbert scheme and the f blowup from the noncommutative viewpoint after van den bergh in doing this we immediately and naturally arrive at the notion of d modules we also find as a byproduct a canonical way to construct a noncommutative resolution at least for a few classes of singularities in positive characteristic

improvement of numerical solution of boundary layer problems by incorporation of asymptotic approximations a method for improvement of the numerical solution of differential equations by incorporation of asymptotic approximations is investigated for a class of singular perturbation problems uniform error estimates are derived for this method when implemented in known difference schemes and applied to linear second order o d e s an improvement by a factor of n 1 can be obtained where is the small parameter andn is the order of the asymptotic approximation for a small amount of extra work numerical experiments are presented

numerical solution of singular perturbation problems via deviating arguments we present a new approach to numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the interval the original problem is divided into outer and inner region problems a terminal boundary condition in the implicit form is derived then the outer region problem is solved as a two point boundary value problem tpbvp and an explicit terminal boundary condition is obtained in turn a new inner region problem is obtained and solved as a tpbvp using the explicit terminal boundary condition the proposed method is iterative on the terminal point of the inner region some numerical examples have been solved to demonstrate the applicability of the method

equivariant ehrhart theory abstract motivated by representation theory and geometry we introduce and develop an equivariant generalization of ehrhart theory the study of lattice points in dilations of lattice polytopes we prove representation theoretic analogues of numerous classical results and give applications to the ehrhart theory of rational polytopes and centrally symmetric polytopes we also recover a character formula of procesi dolgachev lunts and stembridge for the action of a weyl group on the cohomology of a toric variety associated to a root system

the cohomology of the steenrod algebra and representations of the general linear groups let tr k be the algebraic transfer that maps from the coinvariants of certain gl k representations to the cohomology of the steenrod algebra this transfer was defined by w singer as an algebraic version of the geometrical transfer tr k s bv k s s 0 it has been shown that the algebraic transfer is highly nontrivial more precisely that tr k is an isomorphism for k 1 2 3 and that tr k tr k is a homomorphism of algebras in this paper we first recognize the phenomenon that if we start from any degree d and apply sq repeatedly at most k 2 times then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the gl k representations as a consequence every finite sq 0 family in the coinvariants has at most k 2 nonzero elements two applications are exploited the first main theorem is that tr k is not a n isomorphism for k 5 furthermore for every k 5 there are infinitely many degrees in which tr k is not an isomorphism we also show that if tr detects a nonzero element in certain degrees of ker sq 0 then it is not a monomorphism and further for each k tr k is not a monomorphism in infinitely many degrees the second main theorem is that the elements of any sq 0 family in the cohomology of the steenrod algebra except at most its first k 2 elements are either all detected or all not detected by tr k for every k applications of this study to the cases k 4 and 5 show that tr 4 does not detect the three families g d 3 and p and that tr 5 does not detect the family h n 1g n n 1