optimal estimation of a general regression function let e be a convex subset of rd and for each 0 e e let f 0 dt be a probability on the line for any vector an an an2 ann where anl e e let xn1 xnn be independent observations the distribution of xn being f an dt the main results give a method of estimating the unknown regression function an based on a minimum distance recipe under regularity assumptions the proposed estimators are shown in an appropriate framework to be asymptotically normal locally asymptotically minimax and robust the abstract results are illustrated by application to the linear model and to exponential response models in general nothing at all is assumed about the form of the regression function accordingly this forces the limiting normal distributions of the proposed estimators to be located on infinite dimensional linear spaces

integrated local petrov galerkin sinc method for structural mechanics problems an integrated local petrov galerkin sinc method is introduced and applied to static structural mechanics problems the method approximates the highest derivative in the local weak form of the governing equation on a rectangular grid and the lower derivatives and unknown function are found by numerical indenite integration we suggest that the essential boundary conditions may be applied by the traditional penalty method or by a method in which the stiness matrix is reduced to eliminate dependent degrees of freedom we propose and compare the performance of three basis functions in terms of their accuracy and convergence properties for two problems a one dimensional tapered bar and a two dimensional plane stress elasticity problem our results indicate that the present method can provide greater accuracy than the sinc method based on interpolation of highest derivative an integration based collocation method suggested by li and wu li c and wu x numerical solution of dierential equations using sinc method based on

robust estimation via minimum distance methods given a fixed parametric family p it is desired to estimate however due to contaminations of various sorts the data actually collected by the statistician follow a distribution that is close to but possibly distinct from thep s it is proved that under these conditions and in an appropriate asymptotic framework the minimum distance estimators of the cramer von mises type are robust specification of a cramer von mises weight functionh defines a notion of distance each such choice ofh then delineates the kind of contamination possible and leads to an estimator which defends optimally against it when the theory is specialized to location models various choices ofh lead to estimators asymptotically equivalent to such familiar ones as trimmed mean median hodges lehmann estimator and so forth the framework developed herein provides some guidelines for choosing among the possible estimators and suggests that the standard cramer von mises estimator of location is probably as good a robust estimator as any

reductive subgroups of reductive groups in nonzero characteristic abstract let g be a possibly nonconnected reductive linear algebraic group over an algebraically closed field k and let n n the group g acts on gn by simultaneous conjugation let h be a reductive subgroup of g we prove that if k has nonzero characteristic then the natural map of quotient varieties hn h gn g is a finite morphism we use methods introduced by vinberg who proved the same result in characteristic zero as an application we show that if is a finite group then the character variety c g of closed conjugacy classes of representations from to g is finite

completely reducible operatorname sl 2 homomorphisms keywords semisimple groups group scheme homomorphisms frobenius twists optimal homomorphisms geometrically completely reducible images twisted product homomorphisms cocharacters reference ctg article 2007 002doi 10 1090 s0002 9947 07 04289 4 record created on 2008 12 16 modified on 2017 05 12

complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 1 abstract let k be a nonperfect field of characteristic 2 let g be a k split simple algebraic group of type e 6 or g 2 defined over k in this paper we present the first examples of nonabelian non g completely reducible k subgroups of g which are g completely reducible over k our construction is based on that of subgroups of g acting non separably on the unipotent radical of a proper parabolic subgroup of g in our previous work we also present examples with the same property for a non connected reductive group g along the way several general results concerning complete reducibility over nonperfect fields are proved using the recently proved tits center conjecture for spherical buildings in particular we show that under mild conditions a connected k subgroup of g is pseudo reductive if it is g completely reducible over k

restrictions of representations of a surface group to a pair of free subgroups let be the fundamental group of a compact orientable genus m surface and let g be a connected reductive algebraic group over an algebraically closed field of characteristic zero define two free rank m subgroups of by a a 1 a m and b b 1 b m where a 1 a m b 1 b m j 1 m a j b j is the standard presentation of we consider representations of of a and of b into g restriction of representations induces a morphism from c g the variety of closed conjugacy classes of representations of to c a g c b g we prove that if m is greater than the semisimple rank of g then this morphism is dominant and almost all fibers are finite

cocharacter closure and the rational hilbert mumford theorem for a field k let g be a reductive k group and v an affine k variety on which g acts using the notion of cocharacter closed g k orbits in v we prove a rational version of the celebrated hilbert mumford theorem from geometric invariant theory we initiate a study of applications stemming from this rationality tool a number of examples are discussed to illustrate the concept of cocharacter closure and to highlight how it differs from the usual zariski closure

information and asymptotic efficiency in parametric nonparametric models asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems for regular estimates and asymptotic minimax bounds the methods used involve i the notion of a hellinger differentiable root density where part of the differentiation is with respect to the nonparametric part of the model to obtain appropriate scores and ii calculation of the effective score for the real or vector finite dimensional parameter of interest as that component of the score function orthogonal to all nuisance parameter scores perhaps infinite dimensional the resulting asymptotic information for estimation of the parametric component of the model is just 4 times the squared l 2 norm of the effective score a corollary of these results is a simple necessary condition for adaptive estimation adaptation is possible only if the scores for the parameter of interest are orthogonal to the scores for the nuisance function or nonparametric part of the model examples considered include the one sample location model with and without symmetry mixture models the two sample shift model and cox s proportional hazards model

complete reducibility and separability let g be a reductive linear algebraic group over an algebraically closed field of characteristic p 0 a subgroup of g is said to be separable in g if its global and infinitesimal centralizers have the same dimension we study the interaction between the notion of separability and serre s concept of g complete reducibility for subgroups of g the separability hypothesis appears in many general theorems concerning g complete reducibility we demonstrate that many of these results fail without this hypothesis on the other hand we prove that if g is a connected reductive group and p is very good for g then any subgroup of g is separable we deduce that under these hypotheses on g a subgroup h of g is g completely reducible provided the lie algebra of g is semisimple as an h module recently guralnick has proved that if h is a reductive subgroup of g and c is a conjugacy class of g then the intersection of c and g is a finite union of h conjugacy classes for generic p when certain extra hypotheses hold including separability this follows from a well known tangent space argument due to richardson but in general it rests on lusztig s deep result that a connected reductive group has only finitely many unipotent conjugacy classes we show that the analogue of guralnick s result is false if one considers conjugacy classes of n tuples of elements from h for n 1

a normal subgroup of a strongly reductive subgroup is strongly reductive abstract let h be a strongly reductive subgroup of a reductive linear algebraic group g over an algebraically closed field k we prove that any closed normal subgroup n of h is also a strongly reductive subgroup of g if g gl n k then this is a consequence of clifford s theorem from representation theory

on conjugacy classes of maximal subgroups of finite simple groups and a related zeta function we prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of lie type of bounded rank is bounded for exceptional groups this solves a longstanding open problem the proof uses among other tools some methods from geometric invariant theory using this result we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of lie type groups of fixed rank drawing conclusions regarding the behaviour of the corresponding zeta function g s m maxg g m s which appears in many probabilistic applications more specifically we are able to show that for simple groups g and for any fixed real number s 1 g s 0 as g this confirms a conjecture made in 27 we also apply these results to prove the conjecture made in 29 that the symmetric group sn has n o 1 conjugacy classes of primitive maximal subgroups 2000 mathematics classification numbers 20e28 20g15 20d06

numerical analysis of thick plates bending behaviour of a rectangular plate is analysed with the help of a refined higher order theory the theory is based on a higher order displacement model and the three dimensional hooke s laws for plate material giving rise to a more realistic quadratic variation of the transverse shearing strains and linear variation of the transverse normal strain through the plate thickness it is shown that the segmentation method for the numerical analysis of plates simply supported on two opposite edges can be considered to be the most competitive method in terms of efficiency economy reliability and accuracy in such applications

duality and symmetry in mixed integral methods of elastostatics mixed formulations for integral methods of elastostatic analysis of structures are presented both finite and boundary element approaches are considered a governing system exhibiting symmetry is obtained in both cases by preserving duality in the descriptions of equilibrium and compatibility and reciprocity in the elastic constitutive relations the theorems of virtual forces and displacements are interpreted as the energetic representation of static kinematic duality the quadratic programs associated with the governing system are identified as the theorems of minimum potential energy and co energy mathematical programming theory is used to establish conditions for the existence and uniqueness of solutions

linked interpolation for reissner mindlin plate elements part i a simple quadrilateral in most plate elements using the reissner mindlin assumptions the interpolations used for the lateral displacements w and the rotation involve the independent representation of each variable by its nodal values usually with identical interpolations to ensure a higher order of expansion for displacement w its representation is linked in the present paper with both sets of nodal variables conditions necessary for the use of such expansions are established here and the paper shows the development of a linear quadrilateral element q4bl whose performance and robustness are good although it possesses one singularity if only three degrees of freedom are prescribed in part ii we apply the identical formulation to develop a triangular element t3bl which performs equally well and is fully robust

a geometric approach to complete reducibility let g be a connected reductive linear algebraic group we use geometric methods to investigate g completely reducible subgroups of g giving new criteria for g complete reducibility we show that a subgroup of g is g completely reducible if and only if it is strongly reductive in g this allows us to use ideas of r w richardson and hilbert mumford kempf from geometric invariant theory we deduce that a normal subgroup of a g completely reducible subgroup of g is again g completely reducible thereby providing an affirmative answer to a question posed by j p serre and conversely we prove that the normalizer of a g completely reducible subgroup of g is again g completely reducible some rationality questions and applications to the spherical building of g are considered many of our results extend to the case of non connected g

on a question of kulshammer for homomorphisms of algebraic groups abstract let g be a linear algebraic group over an algebraically closed field of characteristic p 0 we show that if h 1 and h 2 are connected subgroups of g such that h 1 and h 2 have a common maximal unipotent subgroup and h 1 r u h 1 and h 2 r u h 2 are semisimple then h 1 and h 2 are g conjugate moreover we show that if h is a semisimple linear algebraic group with maximal unipotent subgroup u then for any algebraic group homomorphism u g there are only finitely many g conjugacy classes of algebraic group homomorphisms h g such that u is g conjugate to this answers an analogue for connected algebraic groups of a question of b kulshammer in kulshammer s original question h is replaced by a finite group and u by a sylow p subgroup of h the answer is then known to be no in general we obtain some results in the general case when h is non connected and has positive dimension along the way we prove existence and conjugacy results for maximal unipotent subgroups of non connected linear algebraic groups when g is reductive we formulate kulshammer s question and related conjugacy problems in terms of the nonabelian 1 cohomology of unipotent radicals of parabolic subgroups of g and we give some applications of this cohomological approach in particular we analyse the case when g is a semisimple group of rank 2

complete reducibility in good characteristic let g be a simple algebraic group of exceptional type over an algebraically closed field of characteristic p 0 a closed subgroup h of g is called g completely reducible g cr if whenever h is contained in a parabolic subgroup p of g it is contained in a levi subgroup of p in this paper we determine the g conjugacy classes of non g cr simple connected subgroups of g when p is good for g for each such subgroup x we determine the action of x on the adjoint module l g and the connected centraliser of x in g as a consequence we classify all non g cr reductive subgroups of g and we additionally show the existence of and classify the following restricted non g cr semisimple subgroups non g cr reductive subgroups with trivial connected centraliser and subgroups which are maximal among reductive subgroups of g

a mixed hybrid finite element model based on orthogonal functions a mixed hybrid formulation for stress finite elements is presented the stresses and the displacements in the domain of the element and the displacements on the boundary are simultaneously and independently approximated using orthogonal functions the stress approximation functions are used as weighting functions in the weighted residual enforcement of the local compatibility and constitutive equations similarly the displacement approximation functions in the domain and on the boundary are used as weighting functions in the weighting residual enforcement of the local equilibrium equation and of the static boundary conditions respectively legendre polynomials and fourier series are used to illustrate the performance of the finite element formulation when applied to elastostatic problems

sub principal homomorphisms in positive characteristic let g be a reductive group over an algebraically closed field of characteristic p and let u g be a unipotent element of order p suppose that p is a good prime for g we show in this paper that there is a homomorphism sl 2 k g whose image contains u this result was first obtained by d testerman j algebra 1995 using case considerations for each type of simple group and using in some cases computer calculations with explicit representatives for the unipotent orbits the proof we give is free of case considerations except in its dependence on the bala carter theorem our construction of generalizes the construction of a principal homomorphism made by j p serre in invent math 1996 in particular is obtained by reduction modulo from a homomorphism of group schemes over a valuation ring in a number field this permits us to show moreover that the weight spaces of a maximal torus of sl 2 k on lie g are the same as in characteristic 0 the existence of a with this property was previously obtained again using case considerations by lawther and testerman memoirs ams 1999 and has been applied in some recent work of g seitz invent math 2000

a hybrid mixed finite element model based on legendre polynomials for reissner mindlin plates abstract a hybrid mixed formulation for reissner mindlin plates is presented the finite element formulation is derived from first principles of mechanics the stress resultants and the displacements in the domain of the element and the displacements on the boundary are simultaneously and independently approximated using legendre polynomials numerical applications are presented to illustrate the performance of the suggested reissner mindlin plate element

involutions and higher order automorphisms of higgs bundle moduli spaces this research was partially supported by the spanish mineco under the icmat severo ochoa grant no sev 2011 0087 and grant no mtm2013 43963 p and by the european commission marie curie irses moduli programme pirses ga 2013 612534

a triangular element based on reissner mindlin plate theory a new triangular plate blending element based on the reissner mindlin theory is developed through a mixed formulation emanating from the hu washizu variational principle a main feature of the formulation is the use of a linear transverse shear interpolation scheme with discrete constraint conditions on the edges

complete reducibility and commuting subgroups let g be a reductive linear algebraic group over an algebraically closed field of characteristic p 0 we study j p serre s notion of g complete reducibility for subgroups of g specifically for a subgroup h and a normal subgroup n of h we look at the relationship between g complete reducibility of n and of h and show that these properties are equivalent if h n is linearly reductive generalizing a result of serre we also study the case when h mn with m a g completely reducible subgroup of g which normalizes n in our principal result we show that if g is connected n and m are connected commuting g completely reducible subgroups of g and p is good for g then h mn is also g completely reducible

complete reducibility and conjugacy classes of tuples in algebraic groups and lie algebras let h be a reductive subgroup of a reductive group g over an algebraically closed field k we consider the action of h on g n the n fold cartesian product of g with itself by simultaneous conjugation we give a purely algebraic characterization of the closed h orbits in g n generalizing work of richardson which treats the case h g this characterization turns out to be a natural generalization of serre s notion of g complete reducibility this concept appears to be new even in characteristic zero we discuss how to extend some key results on g complete reducibility in this framework we also consider some rationality questions

orbit closures and invariants let g be a reductive linear algebraic group h a reductive subgroup of g and x an affine g variety let x h denote the set of fixed points of h in x and n g h the normalizer of h in g in this paper we study the natural map of quotient varieties psi x h x h n g h rightarrow x g induced by the inclusion x h subseteq x we show that given g and h psi x h is a finite morphism for all affine g varieties x if and only if h is a g completely reducible subgroup of g in the sense defined by serre this was proved in characteristic 0 by luna in the 1970s we discuss some applications and give a criterion for psi x h to be an isomorphism we show how to extend some other results in luna s paper to positive characteristic and also prove the following theorem let h and k be reductive subgroups of g then the double coset hgk is closed for generic g in g if and only if h cap gkg 1 is reductive for generic g in g

efficient robust estimates in parametric models let p n an open subset ofr k be a regular parametric model for a sample ofn independent identically distributed observations this paper describes estimates t n n 1 of which are asymptotically efficient under the parametric model and are robust under small deviations from that model in essence the estimates are adaptively modified one step maximum likelihood estimates which adjust themselves according to how well the parametric model appears to fit the data when the fit seems poor t n discounts observations that would have large influence on the value of the usual one step mle the estimates t n are shown to be asymptotically minimax in the hajek lecam sense for a hellinger ball contamination model an alternative construction of robust asymptotically minimax estimates as modified mle s is described for canonical exponential families

conjugacy classes of n tuples in lie algebras and algebraic groups let g be a reductive algebraic group over an algebraically closed field f of characteristic zero and let l g be the lie algebra of g then g acts on g by inner automorphisms and it acts on l g by the adjoint representation by taking the diagonal actions we get actions of g on the spaces of n tuples g n and l g in this paper we will prove a number of geometric properties of the orbits of g on these spaces of n tuples