nonlinear spectral instability of steady state flow of a viscous liquid past a rotating obstacle we show that a steady state solution to the system of equations of a navier stokes flow past a rotating body is nonlinearly unstable if the associated linear operator mathcal l has a part of the spectrum in the half plane lambda in mathbb c mathrm re lambda 0 our result does not follow from known methods mainly because the basic nonlinear operator is not bounded in the same space in which the instability is studied as an auxiliary result of independent interest we also show that the uniform growth bound of the c 0 semigroup mathrm e mathcal l t is equal to the spectral bound of operator mathcal l

the steady motion of a navier stokes liquid around a rigid body let mathcal r be a body moving by prescribed rigid motion in a navier stokes liquid mathcal l that fills the whole space and is subject to given boundary conditions and body force under the assumptions that with respect to a frame mathcal f attached to mathcal r these data are time independent and that their magnitude is not too large we show the existence of one and only one corresponding steady motion of mathcal l with respect to mathcal f such that the velocity field at the generic point x in space decays like x 1 these solutions are physically reasonable in the sense of finn 10 in particular they are unique and satisfy the energy equation among other things this result is relevant in engineering applications involving orientation of particles in viscous liquid 14

the energy equality and regularity results for the stationary navier stokes equations on the exterior of a rotating obstacle abstract consider the stationary navier stokes equations on the exterior of a rotating body which is also moving in the direction of the axis of rotation with constant velocity k e 1 for every external force f div f f l 2 the existence of a weak solution u satisfying finite dirichlet integral i e u l 2 can be obtained by means of the classical galerkin method we first prove that the weak solution u satisfies the additional regularity property u k e 1 l 4 without any condition on f except for f l 2 this regularity result is enough to ensure the validity of the generalized energy equality using these we therefore obtain the uniqueness for weak solutions under some smallness condition on the data

monoids on groups with bn pairs the theory of linear algebraic monoids has been developed in the last few years by l renner and the author 21 361 as a generalization of the theory of linear algebraic groups 16 40 in applications of the theory of linear algebraic groups one is naturally led to consider groups with bnpairs or tits systems 12 4 9 42 431 this class of groups includes in particular the groups of lie type so that the abstract properties of tits systems are useful in studying the conjugacy classes and representations of groups of lie type see 4 9 our interest is in monoids so let m be a monoid such that the group of units g has a bn pair in general g has very little to do with the rest of m in fact for a given g m g could be an arbitrary semigroup so we introduce three axioms which intimately connect the idempotent structure of m with the structure of g we then say that m is a monoid on g the prime example is a connected regular linear algebraic monoid with zero but there are other natural examples for instance renner 37 is developing a theory of monoids of lie type along the lines of steinberg 41 for groups another example arises from the work of smith 39 and cabanes 3 from their work one can associate with certain irreducible modular representations 4 g gl v a set of idempotents es end v then the monoid m generated by 4 g and e is a monoid on 4 g in the examples we have worked out and very likely in general in the situation of linear algebraic monoids there is a natural type map from the finite lattice of conjugacy classes of idempotents g the lattice of f classes into the power set of the dynkin diagram of g the idea of the type map is due to lex renner and is exploited by renner and the author 31 to show in particular that the system of idempotents can be constructed from the type map in the present situation of monoids on g there is again a type map i which characterizes the system of idempotents 139 0021 8693 89 3 00

universal presheaves on group geometries and modular representations abstract we extend the ideas of sheaf homology on buildings m a ronan and s d smith j algebra 96 1985 319 346 to more general geometries with transitive automorphism groups the main technical result is the construction of a universal extension of certain partial sheaves this guarantees a rich supply of sheaves on such geometries computation of zero homology then affords representations of the group we consider various applications of the technique for example to modular irreducibles for sporadic simple groups

steady state navier stokes flows past a rotating body leray solutions are physically reasonable a rigid body fancyscript b moves in a navier stokes liquid fancyscript l filling the whole space outside fancyscript b we assume that when referred to a frame attached to fancyscript b the nonzero velocity of the center of mass and the angular velocity of fancyscript b are constant and that the flow of fancyscript l is steady our main theorem implies that every weak steady state solution in the sense of leray is in fact physically reasonable in the sense of finn for data of arbitrary size such a theorem improves and generalizes an analogous famous result of babenko math ussr sb 20 1 25 1973 obtained in the case 0

1 cohomology of chevalley groups abstract we relate the 1 cohomology of a finite chevalley group to induced modules and finite geometry as a corollary we get the 1 cohomology of most adjoint modules

a maximum product criterion as a tikhonov parameter choice rule for kirsch s factorization method kirsch s factorization method is a fast inversion technique for visualizing the profile of a scatterer from measurements of the far field pattern we present a tikhonov parameter choice approach based on a maximum product criterion mpc which provides a regularization parameter located in the concave part of the l curve on a log log scale the performance of the method is evaluated by comparing our reconstructions with those obtained via the l curve morozov s discrepancy principle and the svd tail numerical results that illustrate the effectiveness of the mpc in reconstruction problems involving both simulated and real data are reported and analyzed

a computer free construction of j4 define a finite simple group j to be of j4 type or simply j4 provided that j contains an involution z withcjz 21 12 3aut mat22 the purpose of this paper is to give the first computer free construction of a group of j4 type in addition we achieve yet another uniqueness proof for groups of j4 type via the simple connectedness of the 2 local geometry of such a group

a geometric presentation of the 26 dimensional module for f4 q abstract the fixed point sheaf of the group f 4 q acting on its 26 dimensional module v is described it is shown that the 0 homology module is isomorphic to v this gives a presentation of v by geometric generators and relations

classical locally finite tits chamber systems of rank 3 a chamber system v 9 j zi is over some index set z is a set of chambers together with partitions in z of v if ze z and c e v then bj is the join of all partitions fijzi e z and d c is the element of fij containing c d c is again a chamber system over j with fij restricted to z we say q is connected if d c furthermore 111 is the rank of v the most important examples of chamber systems are those obtained from groups if g is a group b a subgroup of g and xi ie z are subgroups of g containing b then 5 g b xi icl has as chambers the cosets bg g e g two such cosets bg bh being i adjacent if and only if xi g xih then g acts chamber transitively on with kernel b n bg and is connected if and only if g x 1 i e z furthermore each chamber system with transitive automorphism group g can be obtained in this way by setting b g and xi gd cr a rank 2 chamber system u hi fi is called a classical generalized ma gon mq 3 if is isomorphic to g b pi p where g is an essentially simple rank 2 group of lie type in the sense of cl b is a bore1 subgroup and p pj are the two maximal parabolic subgroups of g containing b here md is given by the relation vu wj wiw g 1 defined by the weyl group of g a rank 2 chamber system is a generalized digon if is isomorphic to v g pin p pi p where pi p are different proper subgroups of g satisfying g pipj pjpi a connected chamber system over z is called a classical tits chamber system if for each ce and i j c z di j c is either a generalized digon or a classical generalized m gon for some mi e n is focally finite if all d c are finite for motivation to consider such classical locally finite tits chamber systems see 15 and 16 if is a classical tits chamber system then the diagram d z is defined in 9 0021 8693 89 3 00

duality for representations of a reductive group over a finite field the purpose of this paper is to construct a duality operation for representations of a reductuve group over a finite field its effect very roughly speaking is to interchange irreducible representations of small degree with ones of large degree for example the unit and steinberg representation at level of characters this operation has been also considered by alvis 11 curtis 2 and kawanaka 4 we shall now fix some notation g will denote a connected reductive group defined over a finite field fq iv s its weyl group and f the set of orbits of the frobenius map on s the subsets of parametrize the classes of parabolic subgroup of g which are defined over fq let 9 be the class corresponding to z c s thus 9 is the class of bore1 subgroups and s g we denote by g the group of f rational points of g and by 9 the set of parabolic subgroups in 9 with are defined over fq similarly if p e 3 we denote by p its group of fq rational points and by u the group of f rational points of its unipotent radical let k be an algebraically closed field of characteristic zero all g modules will be over k let e be a g module for each z c we define

boundedness almost periodicity and stability of certain navier stokes flows in unbounded domains abstract we investigate the existence uniqueness and stability of bounded and almost periodic mild solutions to several navier stokes flow problems in our strategy we propose a general framework for studying the semi linear evolution equations with certain smoothing properties of the linear part and with the local lipschitz continuity of the nonlinear operator our method is based on interpolation functors combined with differential inequalities our abstract results are applied to navier stokes oseen equations describing flows of incompressible viscous fluid passing a translating and rotating obstacle and to navier stokes equations on aperture domains and or in besov spaces

lp lq estimate of the stokes operator and navier stokes flows in the exterior of a rotating obstacle we consider the motion of a viscous fluid filling the whole three dimensional space exterior to a rotating obstacle with constant angular velocity we develop the l p l q estimates and the similar estimates in the lorentz spaces of the stokes semigroup with rotation effect we next apply them to the navier stokes equation to prove the global existence of a unique solution which goes to a stationary flow as t with some definite rates when both the stationary flow and the initial disturbance are sufficiently small in l3 weak l3 space

on local categories of finite groups let g be a finite group over any finite g poset mathcal p we may define a transporter category as the corresponding grothendieck construction transporter categories are generalizations of subgroups of g and we shall demonstrate the finite generation of their cohomology we record a generalized frobenius reciprocity and use it to examine some important quotient categories of transporter categories customarily called local categories

a geometric presentation for the adjoint module of sl3 k abstract let k be a field the points and lines of the projective plane p k 3 determine corresponding subspaces in the space of 3 3 traceless matrices the adjoint module for the group sl 3 k we show that these subspaces conversely determine the module as their direct limit modulo derivations of the field k

further properties of steady state solutions to the navier stokes problem past a three dimensional obstacle we show that the weak formulation of the steady state navier stokes problem in the exterior of a three dimensional compact set closure of a bounded domain corresponding to a nonzero velocity at infinity and subjected to a given body force is equivalent to a nonlinear equation in appropriate banach spaces we thus show that the relevant nonlinear operator enjoys a number of fundamental properties that allow us to derive many significant results for the original problem in particular we prove that the manifold constituted by the pairs u with the nondimensional speed at infinity reynolds number and u weak solution corresponding to and to a given body force f is for generic f a c one dimensional manifold and that for almost any the number of solutions is finite we also show that for any given f in the appropriate function space and any given 0 the corresponding solutions can be controlled by their specification only in a suitable neighborhood i of the boundary the size

strong solutions to the navier stokes equations around a rotating obstacle we study the existence of strong solutions to the three dimensional navier stokes initial boundary value problem in the domain exterior to a rigid body that rotates with constant angular velocity we show that when the initial data u0 are prescribed in an appropriate functional class a strong solution exists at least in some finite time interval moreover the solution exists for all times provided u0 in suitable norm and the magnitude of do not exceed a certain constant depending only on the kinematic viscosity and on the regularity of in this latter case we also show that the velocity field converges to the velocity field of the corresponding steady state solution

pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity we consider a stationary viscous incompressible flow around a translating and rotating body optimal rates of decay are derived for the velocity and its gradient on the basis of a representation formula involving a fundamental solution constructed by r b guenther and e a thomann the fundamental solution of the linearized navier stokes equations for spinning bodies in three spatial dimensions time dependent case j math fluid mech 8 2006 77 98 for a linearized system

a bound for the dimension of ho fv of modules v λi λj for al q abstract the fixed point sheaf of the group a l q acting on the module v for i j is studied it is shown that for i j and i j l l 1 or 1 2 the dimension of the 0 homology module is naturally bounded in dimension by dim v i dim v j

representations of categories and their applications we define for each small category c a category algebra rc over a base ring r and study its representations when c is an ei category we develop a theory of vertices and sources for rc mod which parameterizes the indecomposable rc modules as a main application we use our theory to find formulas for computing higher inverse limits over c

on the geometry of the adjoint representation of a chevalley group abstract we prove that the adjoint module of a chevalley group not of type c l has a presentation by long root subalgebras subject to certain relations determined by the minimal parabolic subgroups

an existence theorem for the navier stokes flow in the exterior of a rotating obstacle we consider the three dimensional navier stokes initial value problem in the exterior of a rotating obstacle it is proved that a unique solution exists locally in time if the initial velocity possesses the regularity l 1 2 this regularity assumption is the same as that in the famous paper of fujita kato an essential step for the proof is the deduction of a certain smoothing property together with estimates near t 0 of the semigroup which is not an analytic one generated by the operator in the space l 2 where stands for the angular velocity of the rotating obstacle and p denotes the projection associated with the helmholtz decomposition

irreducible modules and parabolic subgroups summary it seems desirable from the point of view of finite group theorists to develop further the representation theory of finite chevalley groups in the characteristic of their definition say p by focusing attention on the full array of p local subgroups that is the parabolics rather than the usual single borel or cartan subgroup the following observation may be regarded as a generalization to arbitrary parabolic subgroups of the standard result 3 theorem 39 d for a borel subgroup namely that in an irreducible module the subspace fixed by a maximal unipotent subgroup is 1 dimensional and so affords an irreducible module for a levi complement which is an abelian cartan subgroup it seems natural to state the result first for algebraic groups

spin modules in characteristic 2 abstract some properties of spin modules for orthogonal groups are developed particularly with regard to parabolic subgroups an application is given to a problem in the structure theory of finite simple groups

existence of weak solutions for the motion of rigid bodies in a viscous fluid we study the evolution of a finite number of rigid bodies within a viscous incompressible fluid in a bounded domain of r d d 2 or 3 with dirichlet boundary conditions by introducing an appropriate weak formulation for the complete problem we prove existence of solutions for initial velocities in h 1 0 omega in the absence of collisions solutions exist for all time in dimension 2 whereas in dimension 3 the lifespan of solutions is infinite only for small enough data

sheaf homology and complete reducibility abstract in the study of finite simple groups by means of the geometries provided by their local subgroups problems of structure often reduce to questions about modular representations in finite characteristic of particular interest are modules spanned by fixed points of sylow groups including high weight modules for chevalley groups for which the homology methods of ronan and smith 6 can be very useful this note presents an elementary sufficient condition for splitting of some reducible modules of this type among the applications we generalize a number of results appearing in recent work of timmesfeld 11 which provided the original inspiration for this analysis some of the ideas had appeared in a special case in 10