weakly lefschetz symplectic manifolds for a symplectic manifold the harmonic cohomology of symplectic divisors introduced by donaldson 1996 and of the more general symplectic zero loci introduced by auroux 1997 are compared with that of its ambient space we also study symplectic manifolds satisfying a weakly lefschetz property that is the s lefschetz property in particular we consider the symplectic blow ups cp m of the complex projective space cp m along weakly lefschetz symplectic submanifolds m cp m as an application we construct for each even integer s 2 compact symplectic manifolds which are s lefschetz but not s 1 lefschetz

endotrivial modules for finite groups of lie type a in nondefining characteristic let g be a finite group such that mathop text sl nolimits n q subseteq g subseteq mathop text gl nolimits n q and z be a central subgroup of g in this paper we determine the group t g z consisting of the equivalence classes of endotrivial k g z modules where k is an algebraically closed field of characteristic p such that p does not divide q the results in this paper complete the classification of endotrivial modules for all finite groups of untwisted lie type a initiated earlier by the authors

the discrete artificial boundary condition on a polygonal artificial boundary for the exterior problem of poisson equation by using the direct method of lines the numerical simulation for the exterior problem of poisson equation is considered we introduced a polygonal artificial boundary ce and designed a discrete artificial boundary condition on it by using the direct method of lines then the original problem is reduced to a boundary value problem defined in a bounded computational domain with a polygonal boundary the finite element approximation of this reduced boundary value problem is considered and it is proved that the finite element approximate problem is well posed furthermore numerical examples show that the discrete artificial boundary condition is very eaective and more accurate than the neumann boundary condition which is often used in engineering literatures 1999 elsevier science s a all rights reserved

artificial boundary conditions for two dimensional incompressible viscous flows around an obstacle abstract in this paper we consider numerical simulation of incompressible viscous flow around an obstacle in velocity pressure formulation two horizontal straight line artificial boundaries are introduced and the original flow is approximated by a flow in an infinite channel with slip boundary condition on the wall then two vertical segment artificial boundaries are introduced to limit the channel to a bounded computational domain in the region of the channel between the vertical boundaries and infinity the velocity of the flow is almost a constant vector in which the navier stokes equations can be linearised by oseen equations and thus a general solution can be derived by using separation of variables artificial boundary conditions on the vertical segments are then designed by imposing the continuity of velocity and normal stress therefore the original problem is reduced to a boundary value problem on a bounded computational domain numerical example shows that our artificial boundary conditions are very effective

on an exterior laplace equation problem with robin boundary condition abstract an exterior problem for laplace s equation in r 3 with a robin boundary condition is investigated the particular form of the assumed boundary condition can be considered to be a perturbation of a dirichlet boundary condition the convergence of the solution to the robin problem to that of the corresponding dirichlet problem is proven for sufficiently smooth prescribed boundary functions the proof uses an integral equation method combined with results dealing with singular perturbation problems in addition the robin problem is reformulated as a new boundary integral equation which is shown to be uniquely solvable a constructive method is established for obtaining the solution to this integral equation an example is supplied to illustrate the method given

asymptotic boundary conditions for dissipative waves general theory an outstanding issue in the computational analysis of time dependent problems is the imposition of appropriate radiation boundary conditions at artificial boundaries accurate conditions are developed which are based on the asymptotic analysis of wave propagation over long ranges employing the method of steepest descents dominant wave groups are identified and simple approximations to the dispersion relation are considered in order to derive local boundary operators the existence of a small number of dominant wave groups may be expected for systems with dissipation estimates of the error as a function of domain size are derived under general hypotheses leading to convergence results some practical aspects of the numerical construction of the asymptotic boundary operators are also discussed

the direct method of lines for the problem of infinite elastic foundation we consider the numerical simulation for the problem of infinite elastic foundation a polygonal artificial boundary is introduced and a discrete artificial boundary condition on it is presented by using the direct method of lines then the original problem is reduced to a boundary value problem on a bounded computational domain which is solved by the finite element method in addition we prove an optimal a priori error bound for the displacement in the bounded computational domain finally numerical example shows that the discrete artificial boundary condition given in this paper is very effective and more accurate than neumann boundary condition which is often used in engineering literatures

positive radial symmetric solutions to an exterior elliptic robin boundary value problem and application abstract we study the existence uniqueness and behavior of positive radial symmetric solutions of the exterior boundary value problem u h x f u in r n b 0 r 0 u n a u on and u x 0 as x when a 0 the robin boundary condition has a wrong sign and this makes the analysis of the problem interesting and non trivial we determine a sharp condition on the coefficient a for the existence of positive spherically symmetric solutions of the boundary value problem the important model case f u u 1 7 a 1 2 r 0 and 1 2 r 0 is related to the initial data problem in general relativity for this model case as an example of application of our results we obtain the existence and uniqueness of the conformal factor corresponding to the hamiltonian constraint equation in the case of a single black hole in the radial symmetric regime

non formal co symplectic manifolds we study the formality of the mapping torus of an orientation preserving diffeomorphism of a manifold in particular we give conditions under which a mapping torus has a non zero massey product as an application we prove that there are non formal compact co symplectic manifolds of dimension m and with first betti number b if and only if m 3 and b 2 or m 5 and b 1 explicit examples for each one of these cases are given

non formal homogeneous spaces several large classes of homogeneous spaces are known to be formal in the sense of rational homotopy theory however it seems that far fewer examples of non formal homogeneous spaces are known in this article we provide several construction principles and characterisations for non formal homogeneous spaces which will yield a lot of examples this will enable us to prove that from dimension 72 on such a space can be found in each dimension

an 8 dimensional nonformal simply connected symplectic manifold we answer in the affirmative the question posed by babenko and taimanov 3 on the existence of nonformal simply connected compact symplectic manifolds of dimension 8

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

on endo trivial modules for p solvable groups in this note we will prove a conjecture of carlson et al 1 thus resolving the question left open in 1 and completing the classification of torsion endo trivial modules for p solvable groups namely we will prove that if g is a finite p nilpotent group which contains a non cyclic elementary abelian p subgroup and k is an algebraically closed field of characteristic p then all simple endo trivial kg modules are 1 dimensional in fact we do rather more we prove the analogous result directly in the case that g is p solvable and contains an elementary abelian p subgroup of order p2 carlson mazza and thevenaz had reduced the proof of this result for p solvable g to the p nilpotent case and had proved the result in the solvable case but our method is somewhat different our proof does require the classification of finite simple groups specifically we require the well known fact that the outer automorphism group of a finite simple group of order prime to p has cyclic sylow p subgroups see for example theorem 7 1 2 of gorenstein et al 4 let us recall that a kg module m is endo trivial if m m k n where n is a projective kg module if g is divisible by p then any endo trivial kg module has dimension prime to p the vertex of any indecomposable endo trivial kg module is a sylow p subgroup of g we remark that if m is an endo trivial kg module which is not 1 dimensional then a sylow p subgroup of g acts faithfully on m m and hence acts faithfully on m

construction of nice nilpotent lie groups abstract we illustrate an algorithm to classify nice nilpotent lie algebras of dimension n up to a suitable notion of equivalence applying the algorithm we obtain complete listings for n 9 on every nilpotent lie algebra of dimension 7 we determine the number of inequivalent nice bases which can be 0 1 or 2 we show that any nilpotent lie algebra of dimension n has at most countably many inequivalent nice bases

irreducible specht modules are signed young modules recently donkin defined signed young modules as a simultaneous generalization of young and twisted young modules for the symmetric group we show that in odd characteristic if a specht module s is irreducible then s is a signed young module thus the set of irreducible specht modules coincides with the set of irreducible signed young modules this provides evidence for our conjecture that the signed young modules are precisely the class of indecomposable self dual modules with specht filtrations the theorem is false in characteristic two

numerical simulations of fracture problems by coupling the fem and the direct method of lines in this paper we propose a method for calculating stress intensity factors of plane fracture problems with cracks by coupling the finite element method fem and the direct method of lines we introduce polygonal artificial boundaries and divide the physical domain into two parts a large domain without any crack tips and small polygonal domains surrounding the crack tips we solve the problem defined in the small domains with crack tips by using the direct method of lines and design discrete nonlocal artificial boundary conditions on the polygonal artificial boundaries by imposing the continuity of displacement and normal stress then the original problem is reduced to a boundary value problem defined in a domain without any crack tips the finite element approximation of the reduced problem is considered and we can prove that the finite element approximation is well posed numerical examples and results of a fracture problem with exact solution and two typical fracture problems demonstrate the efficiency and accuracy of the present method

high order boundary conditions and finite elements for infinite domains a finite element method for the solution of linear elliptic problems in infinite domains is proposed the two dimensional laplace helmholtz and modified helmholtz equations outside an obstacle and in a semi infinite strip are considered in detail in the proposed method an artificial boundary b is first introduced to make the computational domain finite then the exact nonlocal dirichlet to neumann dtn boundary condition is derived on b this condition is localized and a sequence of local boundary conditions on b of increasing order is obtained the problem in with a localized dtn boundary condition on b is then solved using the finite element method the numerical stability of the scheme is discussed a hierarchy of special conforming finite elements is developed and used in the layer adjacent to b in conjunction with the local high order boundary condition applied on b an error analysis is given for both nonlocal and local boundary conditions numerical experiments are presented to demonstrate the performance of the method

formality of donaldson submanifolds we introduce the concept of s formal minimal model as an extension of formality we prove that any orientable compact manifold m of dimension 2n or 2n 1 is formal if and only if m is n 1 formal the formality and the hard lefschetz property are studied for the donaldson submanifolds of symplectic manifolds constructed in 13 this study permits us to show an example of a donaldson symplectic submanifold of dimension eight which is formal simply connected and does not satisfy the hard lefschetz theorem

einstein solvmanifolds existence and non existence questions the aim of this paper is to study the problem of which solvable lie groups admit an einstein left invariant metric the space mathcal n of all nilpotent lie brackets on mathbb r n parametrizes a set of n 1 dimensional rank one solvmanifolds s mu mu in mathcal n containing the set of all those which are einstein in that dimension the moment map for the natural gl n action on mathcal n evaluated at mu in mathcal n encodes geometric information on s and suggests the use of strong results from geometric invariant theory for instance the functional on mathcal n whose critical points are precisely the einstein s s is the square norm of this moment map we use a gl n invariant stratification for the space mathcal n and show that there is a strong interplay between the strata and the einstein condition on the solvmanifolds s as an application we obtain criteria to decide whether a given nilpotent lie algebra can be the nilradical of a rank one einstein solvmanifold or not we find several examples of mathbb n graded even 2 step nilpotent lie algebras which are not a classification in the 7 dimensional 6 step case and an existence result for certain 2 step algebras associated to graphs are also given

an open boundary condition for the computation of the steady incompressible navier stokes equations abstract we design a new boundary condition on the stream function at outflow for the steady incompressible navier stokes equations written in the stream function vorticity formulation by using a fourier transform in the direction perpendicular to the flow our open boundary condition is compared to the dirichlet and neumann boundary conditions by computing the flow past an ellipse of excentricity 50 our boundary condition proves to be the most accurate and thus the most time saving

simple endotrivial modules for quasi simple groups we investigate simple endotrivial modules of finite quasi simple groups and classify them in several important cases this is motivated by a recent result of robinson 41 showing that simple endotrivial modules of most groups come from quasi simple groups

simply connected k contact and sasakian manifolds of dimension 7 we construct a compact simply connected 7 dimensional manifold admitting a k contact structure but not a sasakian structure we also study rational homotopy properties of such manifolds proving in particular that a simply connected 7 dimensional sasakian manifold has vanishing cup product h 2 times h 2 rightarrow h 4 and that it is formal if and only if all its triple massey products vanish

symplectic resolutions lefschetz property and formality we introduce a method to resolve a symplectic orbifold m omega into a smooth symplectic manifold m over tilde omega over tilde then we study how the formality and the lefschetz property of m over tilde omega over tilde are compared with that of m omega we also study the formality of the symplectic blow up of m omega along symplectic submanifolds disjoint from the orbifold singularities this allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the lefschetz property but is not formal therefore giving a counter example to a conjecture of babenko and taimanov

on the decomposition numbers of the ree groups f42 q2 in non defining characteristic abstract we compute the l modular decomposition matrices of the simple ree groups f 4 2 q 2 where q 2 2 2 n 1 and n 0 for all primes l 3 up to some entries in the unipotent and one entry in the non unipotent characters we determine the smallest degree of a nontrivial irreducible l modular representation of f 4 2 q 2 for all primes l 3 and also obtain results on the 3 modular decomposition matrices of f 4 2 q 2

the 2 blocks of the covering groups of the symmetric groups abstract let n be a double cover of the finite symmetric group s n of degree n i e n has a central involution z such that n z s n an irreducible character of n is called ordinary or spin according to whether it has z in its kernel or not the purpose of this paper is to determine the distribution of the spin characters of n into 2 blocks the methods applied here are essentially different from those applied to previous questions of this type we also discuss some consequences of our main result for the decomposition numbers an analogue of james well known result for the decomposition numbers of the symmetric groups is proved providing also a generalization of a theorem of benson ben theorem 1 2 in section 1 we present the background for our results and give some preliminaries in section 2 we give an explicit formula for the number of spin characters in a 2 block we also prove a result about the weight of a block containing a given non self associate spin character which will be important for the proof of our theorem on the 2 block distribution of spin characters section 3 presents some fundamental combinatorial concepts used in sections 4 and 5 the theorem concerning the spin characters in a given 2 block is proved in section 4 and in section 5 we present our results on the decomposition numbers

brauer trees for the schur cover of the symmetric group abstract we determine the brauer trees of the faithful blocks of weight 1 of the schur covers of the symmetric group and the alternating group in odd characteristic

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

numerical simulation for the problem of infinite elastic foundation in this paper we consider the numerical simulation for the problem of infinite elastic foundation a half circle artificial boundary is introduced and discrete artificial boundary condition on it is designed by using the direct method of lines then the original problem is reduced to a boundary value problem on a bounded computational domain furthermore finite element approximation of this boundary value problem defined in the bounded domain is considered the numerical results show that the artificial boundary condition given in this paper is very effective and more accurate than the neumann boundary condition which is often used in engineering literatures

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

classification of minimal algebras over any field up to dimension 6 we give a classification of minimal algebras generated in degree 1 defined over any field k of characteristic different from 2 up to dimension 6 this recovers the classification of nilpotent lie algebras over k up to dimension 6 in the case of a field k of characteristic zero we obtain the classification of nilmanifolds of dimension less than or equal to 6 up to k homotopy type finally we determine which rational homotopy types of such nilmanifolds carry a symplectic structure

exact artificial boundary conditions for continuum and discrete elasticity for the continuum and discrete elastic equations we derive exact artificial boundary conditions abcs often referred to as transparent boundary conditions that can be applied at a planar interface below which there are no forces solution of the elasticity equations can then be performed using this interface as an artificial boundary often with greatly reduced computational effort but without loss of accuracy a general solvability requirement is presented for the existence of an artificial boundary operator for discrete systems such as discrete elasticity on an unbounded semi infinite domain the solvability requirement is validated by introducing a sum of exponentials ansatz for the solution below the artificial boundary we also derive a new expression for the total energy for the system involving only the region above the artificial boundary numerical examples are provided to confirm and illustrate the accuracy and effectiveness of the results

infinite domain elements a parametric element is formulated which enables the economic modelling of infinite domain type problmes a typical problem is an opening in a stress field in an infinite medium either in two or three dimensions the strategy is to model around the opening with two or three layers of conventional isoparametric finite elements and surround these with a single layer of infinite domain elements several sample problems has been analysed for circular square and spherical openings in infinite media and the results compared with either theoretical or boundary element solutions which include the infinite boundary in their solution technique copyright

a novel boundary infinite element a new improved infinite element based on mapping is proposed for modelling unbounded domain problems mapping has previously been used in such problems not only for the entire domain but also for the infinite element itself the novelty of the present approach is the simplicity of the mapping and the retention of standard integration abscissae and weights the implementation of these elements is very straightforward and they are ideally suited for the analysis of infinite domain static problems in two and three dimensions

the structure of blocks with a klein four defect group we prove erdmann s conjecture stating that every block with a klein four defect group has a simple module with trivial source and deduce from this that puig s finiteness conjecture holds for source algebras of blocks with a klein four defect group the proof uses the classification of finite simple groups

spin block inclusions abstract in this paper we classify all block equalities and all nontrivial block inclusions for spin blocks of the double covers of the symmetric groups at different odd primes more generally we describe for an odd integer s 1 explicitly when an s block of bar partitions is contained in a t block of bar partitions for t 1 an odd integer or t 4 the question of block equality leads to the study of s t cores in the case of primes they label spin characters which are of defect 0 for different primes and therefore represent a block equality we enumerate these cores and show that there is a unique maximal one

splendid morita equivalences for principal 2 blocks with dihedral defect groups given a dihedral 2 group p of order at least 8 we classify the splendid morita equivalence classes of principal 2 blocks with defect groups isomorphic to p to this end we construct explicit stable equivalences of morita type induced by specific scott modules using brauer indecomposability and gluing methods we then determine when these stable equivalences are actually morita equivalences and hence automatically splendid morita equivalences finally we compute the generalised decomposition numbers in each case

error estimates for the finite element approximation of linear elastic equations in an unbounded domain in this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain the finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one in fact there are a family of nonlocal approximate boundary conditions with increasing accuracy and computational cost and a family of local ones for a given artificial boundary our error estimates show how the errors of the finite element approximations depend on the mesh size the terms used in the approximate artificial boundary condition and the location of the artificial boundary a numerical example for navier equations outside a circle in the plane is presented numerical results demonstrate the performance of our error estimates

discrete dirichlet to neumann maps for unbounded domains abstract it is shown how to construct a discrete counterpart of the dirichlet to neumann dtn map for use on an artificial boundary b introduced in exterior boundary value problems following the idea of deakin and dryden this discrete map provides an approximate non reflecting or absorbing boundary condition for use in formulating a problem in the finite computational domain bounded by b different discrete maps are constructed for use with finite difference and finite element methods in the solution of some simple problems shows that use of the discrete dirichlet to neumann ddtn map yields nearly the same accuracy as that of using the continuous dtn map

on fibrations with formal elliptic fibers abstract we prove that for a fibration of simply connected spaces of finite type f e b with f being positively elliptic and h f q not possessing non trivial derivations of negative degree the base b is formal if and only if the total space e is formal moreover in this case the fibration map is a formal map as a geometric application we show that positive quaternion kahler manifolds are formal and so are their associated twistor fibration maps

intrinsic formality and certain types of algebras in this paper a type of algebra is introduced and studied from a rational homotopy point of view using differential graded lie algebras the main aim of the paper is to establish whether or not such an algebra is the rational cohomology algebra of a unique rational homotopy type of spaces that is in the language of rational homotopy whether or not such an algebra is intrinsically formal examples are given which show that in general this is not so 7 8 and 7 9 however whilst it is true that not all such algebras are intrinsically formal some of them are the main results of this paper show a certain class of these algebras to be intrinsically formal theorem 2 6 1 and a second different type of algebra also to be intrinsically formal theorem 1 5 2 which type of algebra overlaps with the first type in many examples of interest examples are given in 7 1 preliminaries notation and an acknowledgment in d gr mo su it is proven that compact kahler manifolds are formal spaces from the rational homotopy point of view this means that compact kahler manifolds are particularly interesting as examples this paper is concerned with a more algebraic approach than that of d gr mo su the cohomological properties of compact kahler manifolds are abstracted out see 3 1 and algebras satisfying these properties are studied with a particular view to deciding whether or not such algebras are intrinsically formal see 2 1 the main results of this paper are theorems 1 and 2 5 2 and 6 1 many examples are also given in 7 also in 7 examples are given of algebras that are not intrinsically formal and yet satisfy the properties referred to above see 7 8 7 9 rather than give a step by step breakdown of the paper here i have put one or two sentences heading each section these section headings can be read in order at this stage so as to give an overview of the contents of the paper for the fundamentals of rational homotopy theory the basic references are in alphabetical order bou gu gr mo hi q su and su2 this paper will throughout only be concerned with the simply connected with rational homology of finite type case for references which give all the results needed for this received by the editors july 29 1987 1980 mathematics subject classification 1985 revision primary 55p62 55p15 secondary 14m10 53c55

einstein solvmanifolds and the pre einstein derivation an einstein nilradical is a nilpotent lie algebra which can be the nilradical of a metric einstein solvable lie algebra the classification of riemannian einstein solvmanifolds possibly of all noncompact homogeneous einstein spaces can be reduced to determining which nilpotent lie algebras are einstein nilradicals and to finding for every einstein nilradical its einstein metric solvable extension for every nilpotent lie algebra we construct an essentially unique derivation the pre einstein derivation the solvable extension by which may carry an einstein inner product using the pre einstein derivation we then give a variational characterization of einstein nilradicals as an application we prove an easy to check convex geometry condition for a nilpotent lie algebra with a nice basis to be an einstein nilradical and also show that a typical two step nilpotent lie algebra is an einstein nilradical the theory of riemannian homogeneous spaces with an einstein metric splits into three very different cases depending on the sign of the einstein constant the scalar curvature among them the picture is complete only in the ricci flat case by the result of ak every ricci flat homogeneous space is flat the major open conjecture in the case of negative scalar curvature is the alekseevski conjecture al1 asserting that a noncompact einstein homogeneous space admits a simply transitive solvable isometry group this is equivalent to saying that any such space is a solvmanifold a solvable lie group with a left invariant riemannian metric satisfying the einstein condition by a deep result of j lauret la5 any einstein solvmanifold is standard this means that the metric solvable lie algebra s of such a solvmanifold has the following property the orthogonal complement to the derived algebra of s is abelian the systematic study of standard einstein solvmanifolds and the term standard originated from the paper of j heber heb on the lie algebra level all the metric einstein solvable lie algebras can be obtained as the result of the following construction heb la1 la5 lw one starts with the three pieces of data a nilpotent lie algebra n a semisimple derivationof n and an inner product h i n on n with respect to which is symmetric an extension of n byis a solvable lie algebra s rh n as a linear space with adh n the inner product on s is defined by h h ni 0 k hk 2 tr and coincides with the existing one on n the resulting metric solvable lie algebra s h i is einstein provided n is nice and the derivationand the inner product h i n are chosen in the correct way note however that these conditions are expressed by a system of algebraic equations which could hardly be analyzed directly see section 2 metric einstein solvable lie algebras of higher rank with the codimension of the nilradical greater than one having the same nilradical n can be obtained from s via a known procedure by further adjoining to n semisimple derivation commuting with it turns out that the structure of an einstein metric solvable lie algebra is completely encoded in its nilradical in the following sense given a nilpotent lie algebra n there is no more than one possibly none choice ofand of h i n up to conjugation by aut n and scaling which may result in an einstein metric solvable lie algebra s h i definition 1 a nilpotent lie algebra is called an einstein nilradical if it is the nilradical of an einstein metric solvable lie algebra a derivationof an einstein nilradical n and an inner product h i n for which the metric solvable lie algebra s h i is einstein are called an einstein derivation and a nilsoliton inner product respectively

on cartan matrices and lower defect groups for covering groups of symmetric groups we determine the elementary divisors of the cartan matrices of spin p blocks of the covering groups of the symmetric groups when p is an odd prime as a consequence we also compute the determinants of these cartan matrices and in particular we confirm a conjecture by brundan and kleshchev that these determinants depend only on the weight but not on the sign of the block

blow ups and resolutions of strong kahler with torsion metrics abstract on a compact complex manifold we study the behaviour of strong kahler with torsion strong kt structures under small deformations of the complex structure and the problem of extension of a strong kt metric in this context we obtain the analogous result of miyaoka extension theorem studying the blow up of a strong kt manifold at a point or along a complex submanifold we prove that a complex orbifold endowed with a strong kt metric admits a strong kt resolution in this way we obtain new examples of compact simply connected strong kt manifolds

einstein solvmanifolds are standard we study einstein manifolds admitting a transitive solvable lie group of isometries solvmanifolds it is conjectured that these exhaust the class of noncompact homogeneous einstein manifolds j heber has shown that under a simple algebraic condition he calls such a solvmanifold standard einstein solvmanifolds have many remarkable structural and uniqueness properties in this paper we prove that any einstein solvmanifold is standard by applying a stratification procedure adapted from one in geometric invariant theory due to f kirwan

classification of 6 dimensional nilpotent lie algebras over fields of characteristic not 2 abstract nilpotent 6 dimensional lie algebras over any field of characteristic not 2 are classified the proof of this classification is essentially constructive for a given 6 dimensional nilpotent lie algebra l following the steps of the proof it is possible to find a lie algebra m that occurs in the classification and an isomorphism l m in the proof a method due to skjelbred and sund is used along with a method based on grobner bases to find isomorphisms

finite element method with nonlocal boundary condition for solving the nondestructive testing problem of wood moisture content abstract in this paper we use the finite element method fem with nonlocal boundary condition for solving the nondestructive testing problem of wood moisture content based on a planar capacitance sensor model i e the dtn fem for solving the mathematical model which is described by the exterior problems of a class of 3d laplace equation with complicated boundary conditions the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a dirichlet to neumann dtn mapping represented in terms of the fourier expansion series for numerical computation a series of approximate problems with higher accuracy can be derived if one truncates the series term in the variational formulation which is equivalent to the reduced problem the error estimate is presented to show how the error depends on the finite element discretization and the accuracy of the approximate problem based on the numerical results the relationship between the dielectric constant dc of tested wood and the capacitance value of the sensor is discussed finally we applied a least squares fitting method lsfm to reconstruct the wood moisture content wmc from the data measured with a planar capacitance sensor compared with popular statistical methods the hybrid experimental computational method is more convenient and faster and a large number of experiments are avoided the costs of testing are reduced

high order artificial boundary conditions for ideal axisymmetric irrotational flow around 3d obstacles in this paper the numerical solution for ideal irrotational incompressible flow around axisymmetric 3d obstacles is discussed with the artificial boundary method by introducing an artificial boundary we divide the exterior unbounded domain into a bounded part and an unbounded part after setting up a proper artificial boundary condition we get an approximate reduced problem defined on the bounded part both non local and local artificial boundary conditions are designed numerical experiment is also presented and its result demonstrates the effectiveness of these artificial boundary conditions copyright 2002 john wiley sons ltd

the stable category and invertible modules for infinite groups abstract we construct a well behaved stable category of modules for a large class of infinite groups we then consider its picard group which is the group of invertible or endotrivial modules we show how this group can be calculated when the group acts on a tree with finite stabilisers

high order local artificial boundary conditions for problems in unbounded domains in this paper we present error estimates for the finite element approximation of poisson and modified helmholtz equations outside an obstacle or in a semi infinite strip in the plane the finite element approximation is formulated on a bounded domain using a local approximate artificial boundary condition in fact there is a sequence of local approximate boundary conditions for a given artificial boundary our error estimates are based on the mesh size and the location of the artificial boundary the numerical stability and robustness of the method are discussed numerical experiments are presented to demonstrate the performance of the method and our error estimates

the lefschetz property formality and blowing up in symplectic geometry in this paper we study the behaviour of the lefschetz property under the blow up construction we show that it is possible to reduce the dimension of the kernel of the lefschetz map if we blow up along a suitable submanifold satisfying the lefschetz property we use this together with results about massey products to construct compact nonformal symplectic manifolds satisfying the lefschetz property

on the morita frobenius numbers of blocks of finite reductive groups abstract we show that the morita frobenius number of the blocks of the alternating groups the finite groups of lie type in defining characteristic and the ree and suzuki groups is 1 we also show that the morita frobenius number of almost all of the unipotent blocks of the finite groups of lie type in non defining characteristic is 1 and that in the remaining cases it is at most 2

perfect isometries and murnaghan nakayama rules acknowledgement the authors wish to thank the referee for a very careful and precise reading of several earlier versions of this manuscript they are grateful for the number quality and helpfulness of comments and suggestions received

special finite elements for use with high order boundary conditions abstract a hierarchy of special two dimensional finite elements is constructed for use in a layer near a boundary or an interface b the construction of the elements guarantees high order c n regularity along b and c 0 regularity elsewhere all the elements in the hierarchy are proved to be fully conforming the most important finite element in this class is the first order one which provides c 1 regularity on b and c 0 regularity elsewhere it is used here to improve the numerical solution of dirichlet problems by prescribing the tangential derivative of the solution on b in addition to the solution itself this element and higher order ones are also useful near artificial boundaries where high order local non reflecting boundary conditions are used and as transition elements near the interface between a domain governed by a c 0 theory and a domain governed by a c n theory

mixed finite element method and high order local artificial boundary conditions for exterior problems of elliptic equation the mixed finite element method is used to solve the exterior elliptic problem with high order local artificial boundary conditions new unknowns are introduced to reduce the order of the derivatives to two this leads to an equivalent mixed variational problem such that the normal finite element can be used and special finite elements are no longer needed on the adjacent layer of the artificial boundary error estimates are obtained for some local artificial boundary conditions with prescribed order numerical examples are presented and the results demonstrate the effectiveness of this method

simple endotrivial modules for linear unitary and exceptional groups motivated by a recent result of robinson showing that simple endotrivial modules essentially come from quasi simple groups we classify such modules for finite special linear and unitary groups as well as for exceptional groups of lie type our main tool is a lifting result for endotrivial modules obtained in a previous paper which allows us to apply character theoretic methods as one application we prove that the ell rank of quasi simple groups possessing a faithful simple endotrivial module is at most 2 as a second application we complete the proof that principal blocks of finite simple groups cannot have loewy length 4 thus answering a question of koshitani kulshammer and sambale our results also imply a vanishing result for irreducible characters of special linear and unitary groups

methods for parametrizing varieties of lie algebras abstract a real n dimensional anticommutative nonassociative algebra is represented by an element of 2 r n r n for each 2 r n r n there is a unique subset i j k 1 i j n 1 k n so that the structure constant i j k with i j is nonzero if and only if i j k the set of all 2 r n r n with subset is denoted s r these sets stratify 2 r n r n we describe how to smoothly parametrize the lie algebras in s r up to isomorphism for satisfying certain frequently seen hypotheses

on unipotent blocks and their ordinary characters there is a great deal of evidence suggesting that block theory for finite reductive groups should lead to an enrichment of the harish chandra theory for those groups see s 5 1 fs2 p2 3 4 h 6 1 10 an e generalized harish chandra theory for e a positive integer is introduced in bmm i a the context is as follows let g be a connected reductive group defined over lfq with associated frobenius endomorphism f let e 1 an integer when l is an f stable levi subgroup of g one denotes by r and re the lusztig functor and its adjoint a unipotent e pair l l is defined by an e split levi subgroup l of g and a unipotent character z d l 1 one writes li z1 l2 j 2 if and only if l1 c l 2 and zl l2 l 1 2 l 4 0 see 3 l the notion of unipotent e cuspidal pairs and the resulting partition of g g v 1 are defined as in the classical case corresponding to e 1 if f is a prime not dividing q and e is the order of q mod f it seems natural in connection with blocks to ask if the partition of the unipotent characters 8 g v l resulting from e theory coincides with f blocks the blocks of g v which are defined by unipotent characters are called unipoteut f blocks we prove

error estimates for the finite element approximation of problems in unbounded domains in this paper we present error estimates for the finite element approximation of linear elliptic problems in unbounded domains that are outside an obstacle and a semi infinite strip in the plane the finite element approximation is formulated on a bounded domain using a nonlocal approximate artificial boundary condition in fact there is a family of approximate boundary conditions with increasing accuracy and computational cost for a given artificial boundary our error estimates are based on the mesh size the terms used in the approximate artificial boundary condition and the location of the artificial boundary numerical examples for poisson s problem outside a circle and in a semi infinite strip are presented numerical results demonstrate the performance of our error estimates

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