mean value in invexity analysis abstract in this paper a generalization of the mean value theorem is considered in the case of functions defined on an invex set with respect to which is not necessarily connected

criteria for generalized invex monotonicities abstract in this paper under appropriate conditions we establish that i if the gradient of a function is strictly pseudo monotone then the function is strictly pseudo invex ii if the gradient of a function is quasi monotone then the function is quasi invex and iii if the gradient of a function is strong pseudo monotone then the function is strong pseudo invex

characterization of a condition related to a class of preinvex functions in the theory of invex functions a very important role is played by the so called condition c we present a convenient characterization of this condition in terms of starlike sets and projections

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

birational sheets in reductive groups we define the group analogue of birational sheets a construction performed by losev for reductive lie algebras for g semisimple simply connected we describe birational sheets in terms of lusztig spaltenstein induction and we prove that they form a partition of g and that they are unibranch varieties with smooth normalization by means of a local study

complex geometry and the asymptotics of harish chandra modules for real reductive lie groups i let g be a connected semisimple real matrix group it is now apparent that the representation theory of g is intimately connected with the complex geometry of the flag variety 9 by studying appropriate orbit structures on 9 we are naturally led to representation theory in the category of harish chandra modules x or the representation theory of category 9 the jacquet functor j f 9 has proved a useful tool in converting yfw problems into 9 problems which are often more tractable in this paper we advance the philosophy that the complex geometry of 9 associated to yf and 9 interacts in a natural way with the functor j leading to deep new information on the structure of jacquet modules this in turn gives new insight into the structure of certain nilpotent cohomology groups associated to harish chandra modules our techniques are based upon many of the ideas present in the proof of the kazhdan lusztig conjectures and bernstein s proof of the jantzen conjecture

combinatorics of the springer correspondence for classical lie algebras and their duals in characteristic 2 abstract we give a combinatorial description of the springer correspondence for classical lie algebras of type b c or d and their duals in characteristic 2 the combinatorics used here is of the same kind as those appearing in the description of generalized springer correspondence for unipotent case of classical groups by lusztig in odd characteristic and by lusztig and spaltenstein in characteristic 2

context based multiscale classification of document images using wavelet coefficient distributions in this paper an algorithm is developed for segmenting document images into four classes background photograph text and graph features used for classification are based on the distribution patterns of wavelet coefficients in high frequency bands two important attributes of the algorithm are its multiscale nature it classifies an image at different resolutions adaptively enabling accurate classification at class boundaries as well as fast classification overall and its use of accumulated context information for improving classification accuracy

rational rigidity for f4 p abstract we prove the existence of certain rationally rigid triples in f 4 p for good primes p i e p 3 thereby showing that these groups occur as regular galois groups over q t and so also over q we show that these triples give rise to rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic 0

green functions and character sheaves purely combinatorial way using earlier ideas of p hall in the general case they were introduced 3 in terms of l adic cohomology as values of certain virtual representations of r o t is a maximal torus defined over fq at unipotent elements the characters of rg 0 at arbitrary elements were then expressible in a simple form in terms of green functions of g and of smaller groups one drawback of the definition of 3 was that it did not allow one to compute the green functions except in some simple cases another definition of green functions was proposed by springer first in terms of trigonometric sums on the lie algebra and later in geometrical terms 17 this definition was applicable only in large characteristic and in that case it was identified with the definition in 3 by springer 17 and kazhdan 10 in 12 i observed that green functions for gln can be redefined in terms of intersection cohomology this led me to a new definition of green functions

computing green functions in small characteristic abstract let g q be a finite group of lie type over a field with q elements where q is a prime power the green functions of g q as defined by deligne and lusztig are known in almost all cases by work of beynon spaltenstein lusztig und shoji open cases exist for groups of exceptional type e 6 2 e 7 e 8 in small characteristics we propose a general method for dealing with these cases which proceeds by a reduction to the case where q is a prime and then uses computer algebra techniques in this way all open cases in type e 6 2 e 7 are solved as well as at least one particular open case in type e 8

generalized invex monotonicity abstract in this paper the generalized invex monotone functions are defined as an extension of monotone functions a series of sufficient and necessary conditions are also given that relate the generalized invexity of the function with the generalized invex monotonicity of its gradient function this new class of functions will be important in order to characterize the solutions of the variational like inequality problem and mathematical programming problem

on the stack of semistable g bundles over an elliptic curve in a recent paper ben zvi and nadler proved that the induction map from b bundles of degree 0 to semistable g bundles of degree 0 over an elliptic curve is a small map with galois group isomorphic to the weyl group of g we generalize their result to all connected components of text bun g for an arbitrary reductive group g we prove that for every degree i e topological type there exists a unique parabolic subgroup such that any semistable g bundle of this degree has a reduction to it and moreover the induction map is small with galois group the relative weyl group of the levi this provides new examples of simple automorphic sheaves which are constituents of eisenstein sheaves for the trivial local system

generic fibers of the generalized springer resolution of type a abstract it is well known that when the lie algebra is of type a d e the springer fiber above a subregular nilpotent element is described by the dynkin diagram and is called the dynkin curve of the lie algebra on the other hand the closure of the minimal nilpotent orbit is obtained by collapsing the zero section of a cotangent bundle of a projective space p k in this article we are interested in the study of the generalized springer resolution of type a we give a complete description of the generalized springer fiber above a generic singularities showing that it is isomorphic to a dynkin curve or to a projective space

the geometry of generalized steinberg varieties for a reductive algebraic group g the steinberg variety of g is the set of all triples consisting of a unipotent element u in g and two borel subgroups of g that contain u we define generalized steinberg varieties that depend on four parameters and analyze in detail two special cases that turn out to be related to distinguished double coset representatives in the weyl group using one of the two special cases we define a parabolic version of a map from the weyl group to a set of nilpotent orbits of g in lie g defined by joseph and study some of its properties

unipotent classes and special weyl group representations abstract we show that various invariants of a unipotent conjugacy class in a connected semisimple group can be recovered purely in terms of data involving the weyl group

the hesselink stratification of nullcones and base change let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 we give a case free proof of lusztig s conjectures lusztig in transform groups 10 449 487 2005 on so called unipotent pieces this presents a uniform picture of the unipotent elements of g which can be viewed as an extension of the dynkin kostant theory but is valid without restriction on p we also obtain analogous results for the adjoint action of g on its lie algebra mathfrak g and the coadjoint action of g on mathfrak g

relationships between pre invex concepts abstract it is well known that some properties of convex optimization problems especially sufficient conditions for optimality and duality conditions are preserved when convexity is weakened to pre invexity however there are several distinct pre invex concepts b vex b pre invex pre univex p r pre invex and these are not equivalent the pre invexity concepts mentioned above have also some properties which are characteristic and invariant for functions belonging to these classes of functions

boundary detection by constrained optimization a statistical framework is used for finding boundaries and for partitioning scenes into homogeneous regions the model is a joint probability distribution for the array of pixel gray levels and an array of labels in boundary finding the labels are binary zero or one representing the absence or presence of boundary elements in partitioning the label values are generic two labels are the same when the corresponding scene locations are considered to belong to the same region the distribution incorporates a measure of disparity between certain spatial features of block pairs of pixel gray levels using the kolmogorov smirnov nonparametric measures of difference between the distributions of these features the number of model parameters is minimized by forbidding label configurations which are assigned probability zero the maximum a posteriori estimator of boundary placements and partitionings is examined the forbidden states introduce constraints into the calculation of these configurations stochastic relaxation methods are extended to accommodate constrained optimization

unitary equivalences for reductive p adic groups we establish a transfer of unitarity for a bernstein component of the category of smooth representations of a reductive p adic group to the associated hecke algebra in the framework of the theory of types whenever the hecke algebra is an affine hecke algebra with geometric parameters in the sense of lusztig possibly extended by a group of automorphisms of the root datum it is known that there is a large class of such examples detailed in the paper as a consequence we establish relations between the unitary duals of different groups in the spirit of endoscopy

macdonald functions associated to complex reflection groups let w be the complex reflection group sn z ez n in the author s previous paper j algebra 245 2001 650 694 hall littlewood functions associated to w were introduced in the special case where w is a weyl group of type bn they are closely related to green polynomials of finite classical groups in this paper we introduce a two variables version of the above hall littlewood functions as a generalization of macdonald functions associated to symmetric groups a generalization of macdonald operators is also constructed and we characterize such functions by making use of macdonald operators assuming a certain conjecture

on special pieces the springer correspondence and unipotent characters let g be a connected reductive algebraic group over the algebraic closure of a finite field bbb f q of good characteristic in this paper we demonstrate a remarkable compatibility between the springer correspondence for g and the parametrization of unipotent characters of g bbb f q in particular we show that in a suitable sense large portions of these two assignments in fact coincide this extends earlier work of lusztig on springer representations within special pieces of the unipotent variety

hamiltonian reduction and nearby cycles for mirabolic d modules abstract we study holonomic d modules on sl n c c n called mirabolic modules analogous to lusztig s character sheaves we describe the supports of simple mirabolic modules we show that a mirabolic module is killed by the functor of hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus we introduce an analogue of verdier s specialization functor for representations of cherednik algebras which agrees on category o with the restriction functor of bezrukavnikov and etingof in type a we also consider a verdier specialization functor on mirabolic d modules we show that hamiltonian reduction intertwines specialization functors on mirabolic d modules with the corresponding functors on representations of the cherednik algebra this allows us to apply known hodge theoretic purity results for nearby cycles in the setting considered by bezrukavnikov and etingof

characterizations of character sheaves for complex reductive algebraic groups character sheaves are geometric objects associated to a reductive algebraic group they are closely related to irreducible characters of finite chevalley groups the existence of character sheaves was conjectured by g lusztig in the book characters of reductive groups over finite fields 1984 there he described what one should expect character sheaves to be for complex reductive groups a definition of character sheaves for an arbitrary characteristic quite different from this description was given by him later 1985 it is shown in this thesis in particular that the original 1984 lusztig s suggestion for describing character sheaves is equivalent to the later definition we use alternative characterizations of character sheaves for a connected complex reductive algebraic group g that are obtained in this thesis and also the one conjectured by g lusztig and g laumon that was obtained in 1988 by v ginsburg and independently by i mirkovic and k vilonen these characterizations involve microsupport and microlocalization of a sheaf a microsupport ss cal f of a complex cal f of sheaves on g is a closed conic subset of t sp g g times bf g it tells us about singularities of cal f a microlocalization mu sb m cal f of cal f along a submanifold m of g is a complex of sheaves on the conormal bundle t sbsp m g which allows us to analyze cal f on the infinitesimally small neighbourhood of m theorem let g be a complex connected reductive algebraic group cal f be a g equivariant for conjugation irreducible perverse sheaf on g let cal n be the nilpotent cone of the lie algebra g of g s may denote either g or the set of all semisimple elements of g the following are equivalent i cal f is a character sheaf ii ss cal f vert sb s subset s times cal n iii supp mu sb cal o cal f subset cal o times cal n where cal o subset s is a conjugacy class iv supp mu sb x cal f subset cal n x in s characterization ii when s g is the one given by ginsburg mirkovic and vilonen characterization iii with s being the set of semisimple elements was conjectured by j bernstein for arbitrary characteristic this kind of characterizations essentially simplifies the development and our understanding of the theory currently there is still no such description for positive characteristic an important intermediate result is the description of the microsupport of a doubc constructible sheaf cal f on a manifold x in terms of microlocalizations to the points of this manifold ss cal f overline bigcup limits sb x in x rm supp mu sb x cal f

on unipotent supports of reductive groups with a disconnected centre let g be a simple algebraic group defined over a finite field of good characteristic with associated frobenius endomorphism f in this article we extend an observation of lusztig which gives a numerical relationship between an ordinary character of gf and its unipotent support to the case where z g is disconnected we then use this observation in some applications to the ordinary character theory of gf 2013 elsevier inc

text extraction and document image segmentation using matched wavelets and mrf model in this paper we have proposed a novel scheme for the extraction of textual areas of an image using globally matched wavelet filters a clustering based technique has been devised for estimating globally matched wavelet filters using a collection of groundtruth images we have extended our text extraction scheme for the segmentation of document images into text background and picture components which include graphics and continuous tone images multiple two class fisher classifiers have been used for this purpose we also exploit contextual information by using a markov random field formulation based pixel labeling scheme for refinement of the segmentation results experimental results have established effectiveness of our approach

green functions associated to complex reflection groups abstract green functions of classical groups are determined by the data from weyl groups and by certain combinatorial objects called symbols generalizing this we define green functions associated to complex reflection groups g e 1 n and study their combinatorial properties we construct hall littlewood functions and schur functions in our scheme and show that such green functions are obtained as a transition matrix between those two symmetric functions as in the case of gln

remarks on computing irreducible characters 0 1 let g be a connected reductive algebraic group defined over a finite field fq and let g fq be the finite group of all fq rational points of g we would like to present here a strategy for computing the character table of g fq under the assumption that p the characteristic of fq is sufficiently large we can assume that g has a simply connected derived group indeed in the general case we can find a connected reductive group g with simply connected derived group and a surjective homomorphism f g g whose kernel is a central torus such that g and f are defined over fq then the character table

g pre invex functions in mathematical programming in the present paper we introduce the concept of g pre invex functions with respect to h defined on an invex set with respect to h these function unify the concepts of nondifferentiable convexity pre invexity and r pre invexity furthermore relationships of g pre invex functions to various introduced earlier pre invexity concepts are also discussed some geometric properties of this class of functions are also derived finally optimality results are established for optimization problems under appropriate g pre invexity conditions

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

vanishing of odd dimensional intersection cohomology ii for a variety where a connected linear algebraic group acts with only finitely many orbits each of which admits an attractive slice we show that the stratification by orbits is perfect for equivariant intersection cohomology with respect to any equivariant local system this applies to provide a relationship between the vanishing of the odd dimensional intersection cohomology sheaves and of the odd dimensional global intersection cohomology groups for example we show that odd dimensional intersection cohomology sheaves and global intersection cohomology groups vanish for all complex spherical varieties

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

on unipotent and nilpotent pieces for classical groups abstract we show that the definition of unipotent resp nilpotent pieces for classical groups given by lusztig resp lusztig and the author coincides with the combinatorial definition using closure relations on unipotent classes resp nilpotent orbits moreover we give a closed formula for a map from the set of unipotent classes resp nilpotent orbits in characteristic 2 to the set of unipotent classes in characteristic 0 such that the fibers are the unipotent resp nilpotent pieces

generalized green functions associated to complex reflection groups abstract in this paper we consider the set of r symbols in a full generality we construct hall littlewood functions and kostka functions associated to those r symbols we also discuss a multi parameter version of those functions we show that there exists a general algorithm of computing the multi parameter kostka functions as an application we show that the generalized green functions of symplectic groups can be described combinatorially in terms of our one parameter kostka functions

nilpotent orbits in classical lie algebras over f2n and the springer correspondence we give the number of nilpotent orbits in the lie algebras of orthogonal groups under the adjoint action of the groups over f2n let g be an adjoint algebraic group of type b c or d defined over an algebraically closed field of characteristic 2 we construct the springer correspondence for the nilpotent variety in the lie algebra of g

characteristic cycles for the loop grassmannian and nilpotent orbits x which is a linear combination of closures of conormal bundles to submanifolds of x intuitively the microlocal multiplicities c me asurethesingularity of at in settings related to representation theory a group g acts on x is g equivariant and the microlocal multiplicities play a significant but only partially understood role in representation theory see ro sv abv and kasa e g in this paper we compute microlocal multiplicities for certain cases of interest in representation theory let g be a connected reductive group with loop group lg and let p bethesubgroup of lg consisting of loops with positive fourier coefficients then p orbits on theloop grassmannian lg p correspond to the irreducible representations l of thedual re ductivegroup g for dominant weights and of a torus of

on invex sets and preinvex functions abstract in this paper we consider the class of the invex function introduced by hanson we show that under certain condition an invex function defined on an invex set a is preinvex on a similarly a quasiinvex function defined on an invex set a is pre quasiinvex

modular generalized springer correspondence iii exceptional groups we complete the construction of the modular generalized springer correspondence for an arbitrary connected reductive group with a uniform proof of the disjointness of induction series that avoids the case by case arguments for classical groups used in previous papers in the series we show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the sylow subgroups of the weyl group under some assumptions we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character we also provide tables and other information on the modular generalized springer correspondence for quasi simple groups of exceptional type including a complete classification of cuspidal pairs in the case of good characteristic and a full determination of the correspondence in type g 2

computational proof of the mackey formula for q 2 let g be a connected reductive group defined over a finite field with q elements we prove that the mackey formula for the lusztig induction and restriction holds in g whenever q 2 or g does not have a component of type e

nilpotent coadjoint orbits in small characteristic abstract we show that the numbers of nilpotent coadjoint orbits in the dual of exceptional lie algebra g 2 in characteristic 3 and in the dual of exceptional lie algebra f 4 in characteristic 2 are finite we determine the closure relation among nilpotent coadjoint orbits in the dual of lie algebras of type b c f 4 in characteristic 2 and in the dual of lie algebra of type g 2 in characteristic 3 in each case we give an explicit description of the nilpotent pieces in the dual defined in 2 which are in general unions of nilpotent coadjoint orbits coincide with the earlier case by case definition in 11 26 in the case of classical groups and have nice properties independent of the characteristic of the base field this completes the classification of nilpotent coadjoint orbits in the dual of lie algebras of reductive algebraic groups and the determination of closure relation among such orbits in all characteristic

produit scalaire elliptique abstract suppose that w is a weyl group let c w be a space of functions on w with complex values invariant under conjugation we can define an elliptic scalar product on c w it is a natural ingredient to the representation theory of p adic reductive groups let g be a reductive group over the algebraic closure of a finite field the generalized springer correspondence gives a bijection between two sets the set of pairs u e where u is an unipotent orbit of g and e is a g equivariant irreducible local system on u the disjoint union of the sets of irreducible representations of certain weyl groups related to g using kazhdan lusztig polynomials we modify the generalized springer correspondence by the modified correspondence a pair u e as above maps to a representation of a certain weyl group and this representation is in general reducible there is no simple formula that relates the elliptic scalar product and the generalized springer correspondence but a simple formula does exist and we prove it that relates the elliptic scalar product and the modified generalized springer correspondence our result is in fact a corollary of a theorem of lusztig on the restriction of character sheaves to the unipotent variety

semistrictly preinvex functions a new class of functions termed semistrictly preinvex functions is introduced in this paper some properties of semistrictly preinvex functions are discussed in particular the relationship between semistrictly preinvex functions and preinvex functions is investigated some characterization properties of semistrictly preinvex functions are derived under a lower semicontinuous condition and a gradient property of semistrictly preinvex functions is obtained

pre invex functions in multiple objective optimization and called such functions convexlike if s is a convex set and if f is a convex function then clearly f is convexlike elster and nehse obtained a saddlepoint optimality condition for convexlike mathematical programs hayashi and komiya 9 also considered convexlike functions and developed a gordan type theorem of the alternative involving convexlike functions and in addition considered lagrangian duality for convexlike programs hanson 7 considered differentiable functions f s r for which there exists an n dimensional vector function q q u such that for all x u e s

text segmentation for mrc document compression the mixed raster content mrc standard itu t t 44 specifies a framework for document compression which can dramatically improve the compression quality tradeoff as compared to traditional lossy image compression algorithms the key to mrc compression is the separation of the document into foreground and background layers represented as a binary mask therefore the resulting quality and compression ratio of a mrc document encoder is highly dependent upon the segmentation algorithm used to compute the binary mask in this paper we propose a novel multiscale segmentation scheme for mrc document encoding based upon the sequential application of two algorithms the first algorithm cost optimized segmentation cos is a blockwise segmentation algorithm formulated in a global cost optimization framework the second algorithm connected component classification ccc refines the initial segmentation by classifying feature vectors of connected components using an markov random field mrf model the combined cos ccc segmentation algorithms are then incorporated into a multiscale framework in order to improve the segmentation accuracy of text with varying size in comparisons to state of the art commercial mrc products and selected segmentation algorithms in the literature we show that the new algorithm achieves greater accuracy of text detection but with a lower false detection rate of nontext features we also demonstrate that the proposed segmentation algorithm can improve the quality of decoded documents while simultaneously lowering the bit rate

on properties of preinvex functions abstract we consider in this paper the class of preinvex functions introduced by t weir and co workers 1988 j math anal appl 136 29 38 1988 bull austral math soc 38 177 189 under semi continuity conditions a determination of the satisfaction of preinvexity for a function can be achieved via an intermediate point preinvexity check a characterization of a preinvex function in terms of its relationship with an intermediate point preinvexity and prequasi invexity is provided

a vanishing theorem for dirac cohomology of standard modules abstract this paper studies the dirac cohomology of standard modules in the setting of graded hecke algebras with geometric parameters we prove that the dirac cohomology of a standard module vanishes if and only if the module is not twisted elliptic tempered the proof makes use of two deep results one is some structural information from the generalized springer correspondence obtained by s kato and lusztig another one is a computation of the dirac cohomology of tempered modules by barbasch ciubotaru trapa and ciubotaru we apply our result to compute the dirac cohomology of ladder representations for type a n for each of such representations with non zero dirac cohomology we associate to a canonical weyl group representation we use the dirac cohomology to conclude that such representations appear with multiplicity one