homotopy types of line arrangements we prove that the complement of a real affine line arrangement inc2 is homotopy equivalent to the canonical 2 complex associated with randell s presentation of the fundamental group this provides a much smaller model for the homotopy type of the complement of a real affine 2 or central 3 arrangement than the salvetti complex and its cousins as an application we prove that these exist infinitely many pairs of central arrangements inc3 with different underlying matroids whose complements are homotopy equivalent we also show that two real 3 arrangements whose oriented matroids are connected by a sequence of flips are homotopy equivalent

ringel duality for perverse sheaves on hypertoric varieties abstract motivated by the polynomial representation theory of the general linear group and the theory of symplectic singularities we study a category of perverse sheaves with coefficients in a field k on any affine unimodular hypertoric variety m our main result is that this is a highest weight category whose ringel dual is the corresponding category for the gale dual hypertoric variety m on the way to proving our main result we confirm a conjecture of finkelberg kubrak in the case of hypertoric varieties we also show that our category is equivalent to representations of a combinatorially defined algebra recently introduced in a related paper

the ring structure on the cohomology of coordinate subspace arrangements every simplicial complex delta subset2 n on the vertex set n 1 ldots n defines a real resp complex arrangement of coordinate subspaces in mathbb r n resp mathbb c n via the correspondence delta ni sigma mapsto rm span e i i in sigma the linear structure of the cohomology of the complement of such an arrangement is explicitly given in terms of the combinatorics of delta and its links by the goresky macpherson formula here we derive by combinatorial means the ring structure on the integral cohomology in terms of data of delta we provide a non trivial example of different cohomology rings in the real and complex case furthermore we give an example of a coordinate arrangement that yields non trivial multiplication of torsion elements

combinatorial stratification of complex arrangements we present a method for discretizing complex hyperplane arrange ments by encoding their topology into a finite partially ordered set of sign vectors this is used in the following ways 1 a general method is given for constructing regular cell complexes having the homotopy type of the complement of the arrangement 2 for the case of complexified arrangements this specializes to the con struction of salvetti s we study the combinatorial structure of complexified arrangements and the salvetti complex in some detail 3 this general method simultaneously produces cell decompositions of the singularity link this link is shown to have the homotopy type of a wedge of d spheres for arrangements in cd d 4 4 the homology of the link and the cohomology of the complement are computed in terms of explicit bases which are matched by alexander duality this gives a new more elementary and more generally valid proof for the brieskorn orlik solomon theorem and some related results 5 our setup leads to a more general notion of 2 pseudoarrangements which can be thought of as topologically deformed complex arrangements re taining only the essential topological and combinatorial structure we show that all of the above remains true in this generality except for the sign patterns of the orlik solomon relations department of mathematics royal institute of technology s10044 stockholm sweden e mail address andersb nada kth se institut fur mathematik universitat augsburg d 8900 augsburg germany current address institut mittag leffler auravagen 17 s 18262 djursholm sweden address after 1 april 1992 konrad zuse zentrum fur informationstechnik berlin heilbronner str 10 w 1000 berlin 31 germany e mail addresses ziegler ml kva se ziegler zib berlin de this content downloaded from 157 55 39 211 on fri 24 jun 2016 06 44 45 utc all use subject to http about jstor org terms

equivariant constructible koszul duality for dual toric varieties abstract for affine toric varieties x and x defined by dual cones we define an equivalence of categories between mixed versions of the equivariant derived category d t b x and the derived category of sheaves on x which are locally constant with unipotent monodromy on each orbit this equivalence satisfies the koszul duality formalism of beilinson ginzburg and soergel

on a complex beam beam interaction model with random forcing abstract in recent years many studies have been devoted to complex differential equations cde which appear in very important applications in physics and engineering this paper aims to investigate one such cde containing a random forcing term z o 2 z e 2 g z e 2 f z z p o t n t where z t x t iy t a complex function i 1 n t is a broad band process with zero mean and e2 and small real parameters in particular we use eq to model the interaction between two colliding beams in particle accelerators setting g z z f z z z z 2 and p o t cos o t and extend the work we had started in an earlier publication mahmoud physica a 216 1995 445 we apply the stochastic averaging method to derive a fokker planck kolmogorov equation for this equation and obtain analytically the exact stationary probability density function and the first and second moments in the amplitude of the solutions numerical simulations are carried out to compare with the theoretical ones and excellent agreement is found

subspace arrangements over finite fields cohomological and enumerative aspects abstract the enumeration of points on or off the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function we discuss the basic relations between these two points of view counting points is also related to the l adic cohomology of the arrangement as a variety we describe the eigenvalues of the frobenius map acting on this cohomology which corresponds to a finer decomposition of the zeta function the l adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data finally it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

nonlinear proximal point algorithms using bregman functions with applications to convex programming a bregman function is a strictly convex differentiable function that induces a well behaved distance measure or d function on euclidean space this paper shows that for every bregman function there exists a nonlinear version of the proximal point algorithm and presents an accompanying convergence theory applying this generalization of the proximal point algorithm to convex programming one obtains the d function proximal minimization algorithm of censor and zenios and a wide variety of new multiplier methods these multiplier methods are different from those studied by kort and bertsekas and include nonquadratic variations on the proximal method of multipliers

the polyhedral product functor a method of decomposition for moment angle complexes arrangements and related spaces this article gives a natural decomposition of the suspension of generalized moment angle complexes or partial product spaces which arise as polyhedral product functors described below the geometrical decomposition presented here provides structure for the stable homotopy type of these spaces including spaces appearing in work of goresky macpherson concerning complements of certain subspace arrangements as well as davis januszkiewicz and buchstaber panov concerning moment angle complexes since the stable decompositions here are geometric they provide corresponding homological decompositions for generalized moment angle complexes for any homology theory

approximation solvability of a class of nonlinear set valued variational inclusions involving a η monotone mappings abstract a new class of nonlinear set valued variational inclusions involving a monotone mappings in a hilbert space setting is introduced and then based on the generalized resolvent operator technique associated with a monotonicity the existence and approximation solvability of solutions using an iterative algorithm is investigated

on the difference between real and complex arrangements if b is an arrangement of linear complex hyperplanes in c d then the following can be constructed from knowledge of its intersection lattice a the cohomology groups of the complement br b the cohomology algebra of the complement os c the fundamental group of the complement if d 2 d the singularity link up to homeomorphism if d 3 e the singularity link up to homotopy type zz if bis more generally a 2 arrangement in ir 2d an arrangement of real subspaces of codimension 2 with even dimensional intersections then the intersection lattice still determines a the cohomology groups of the complement gm and e the homotopy type of the singularity link zz we show however that for 2 arrangements the data b c and d are not deter mined by the intersection lattice they require the knowledge of extra information on sign patterns which can be computed as determinants of linear relations or equiva lently as linking coefficients in the sense of knot theory

coincidences between homological densities predicted by arithmetic abstract motivated by analogies with basic density theorems in analytic number theory we introduce a notion and variations of the homological density of one space in another we use weil s number field function field analogy to predict coincidences for limiting homological densities of various sequences z n d 1 d m x of spaces of 0 cycles on manifolds x the main theorem in this paper is that these topological predictions which seem strange from a purely topological viewpoint are indeed true one obstacle to proving such a theorem is the combinatorial complexity of all possible collisions of points this problem does not arise in the simplest and classical case m n 1 2 of configuration spaces to overcome this obstacle we apply the bjorner wachs theory of lexicographic shellability from algebraic combinatorics

new approach to the η proximal point algorithm and nonlinear variational inclusion problems abstract a new approach using the over relaxed proximal point algorithm in the context of solving a class of inclusion problems based on the notion of maximal monotonicity is developed and examined convergence analysis seems to be reasonable and finally some specializations are included furthermore the model developed in this communication seems to be appropriate to the yosida approximation in the sense that it can be applied to first order evolution equations inclusions as well

linear decision trees subspace arrangements and mobius functions let p be a set in rn given a point x we want to test if x e p our model of computation is a linear decision tree a rooted ternary tree t where with each interior node v is associated a linear function lv x l j ajxj b and the three edges connecting v to its descendants are labelled starting from the root we move down the tree at each internal node v we check whether iv x 0 and follow the appropriately labelled edge leaves are labelled yes and no and arriving at a leaf we read off the answer to the question is x e p linear decision trees are sometimes surprisingly powerful devices we mention here the result that some np complete problems like the knapsack problem have polynomial depth linear decision trees mh lower bound results on the size or depth of linear decision trees usually depend on counting the number of connected components of p or r n p this method was introduced by dobkin and lipton dl and further developed by steele and yao sy ben or bo and others in this paper we describe topological methods that can be applied to obtain lower bounds also if these sets are connected the basic idea of this paper was first explored in bly the euler characteristic of p or r n p was used to estimate the number of leaves in a linear decision tree for p here we improve and extend the topological methods of bly in particular we show that the euler characteristic can be replaced by the sum of the betti numbers of r n p in bl y the method was applied to the case when p is a subspace arrangement i e a union of a finite number of affine subspaces the euler characteristic of the union is essentially the mobius function of the intersection lattice of these subspaces in this paper using a formula of goresky and macpherson gm we work with the betti numbers of the complement of this union and thereby obtain better bounds the motivating problem in bly was the k equal problem given n real numbers xl xn decide if some k of them are equal it was shown that ev ry linear decision tree for the k equal problem has size at least 2njk 0 n and consequently depth at least q nlog 2njk which is best possible we

mixed equilibrium problems sensitivity analysis and algorithmic aspect abstract the aim of this paper is twofold first it is to extend the sensitivity analysis framework developed recently for variational inequalities to mixed equilibrium problems the second is to propose iterative methods for solving this kind of problems in the process we establish an equivalence between an extended version of wiener hopf equations and the given problems relying on a generalization of the yosida approximation notion our results generalize results obtained for optimization variational inequalities complementarity problems and problems of nash equilibria

principal response of duffing oscillator to combined deterministic and narrow band random parametric excitation abstract the principal resonance of a duffing oscillator to combined deterministic and narrow band random parametric excitations is investigated in particular the case in which the parametric terms share close frequencies is examined the method of multiple scales is used to determine the equations of modulation of amplitude and phase the behavior stability and bifurcation of steady state response are studied by means of qualitative analyses jumps are shown to occur if the random excitation is small the effects of damping detuning and magnitudes of deterministic and narrow band parametric excitations are analyzed the theoretical analyses are verified by numerical results

a proximal point algorithm for finding a common zero of a finite family of maximal monotone operators in the presence of computational errors abstract in a hilbert space we study the convergence of a proximal point method to a common zero of a finite family of maximal monotone operators under the presence of computational errors most results known in the literature establish the convergence of proximal point methods when computational errors are summable in the present paper the convergence of the method is established for nonsummable computational errors we show that the proximal point method generates a good approximate solution if the sequence of computational errors is bounded from above by a constant

the extragradient method for finding a common solution of a finite family of variational inequalities and a finite family of fixed point problems in the presence of computational errors abstract in a hilbert space we study the convergence of the subgradient method to a common solution of a finite family of variational inequalities and of a finite family of fixed point problems under the presence of computational errors most results known in the literature establish the convergence of algorithms when computational errors are summable in the present paper the convergence of the subgradient method is established for nonsummable computational errors we show that the subgradient method generates a good approximate solution if the sequence of computational errors is bounded from above by a constant

small rational model of subspace complement this paper concerns the rational cohomology ring of the complement m of a complex subspace arrangement we start with the de concini procesi differential graded algebra that is a rational model for m inside it we find a much smaller subalgebra d quasi isomorphic to the whole algebra d is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice l whence connecting the de concini procesi model with the goresky macpherson formula for the additive structure of h m the algebra d has a natural integral version that is a good candidate for an integral model of m if the rational local homology of l can be computed explicitly we obtain an explicit presentation of the ring h m q for example this is done for the cases where l is a geometric lattice and where m is a k equal manifold

applications of splitting algorithm to decomposition in convex programming and variational inequalities recently han and lou proposed a highly parallelizable decomposition algorithm for minimizing a strongly convex cost over the intersection of closed convex sets it is shown that their algorithm is in fact a special case of a splitting algorithm analyzed by gabay for finding a zero of the sum of two maximal monotone operators gabay s convergence analysis for the splitting algorithm is sharpened and new applications of this algorithm to variational inequalities convex programming and the solution of linear complementarily problems are proposed for convex programs with a certain separable structure a multiplier method that is closely related to the alternating direction method of multipliers of gabay mercier and of glowinski marrocco but which uses both ordinary and augmented lagrangians is obtained

note on b splines wavelet scaling functions and gabor frames let g be a continuous compactly supported function on such that the integer translates of g constitute a partition of unity we show that the gabor system g a b with window g and time shift and frequency shift parameters a b 0 has no lower frame bound larger than 0 if b 2 3 and a 0 in particular g a b is not a gabor frame if g is a continuous compactly supported wavelet scaling function and if b 2 3 and a 0 we give an example for our result for the case that g b sub 1 the triangle function supported by 1 1 by showing pictures of the canonical dual corresponding to g a b where ab 1 4 and b crosses the lines n 2 3

smooth methods of multipliers for complementarity problems this paper describes several methods for solving nonlinear complementarity problems a general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem obtaining primal dual and primal dual formulations we derive bregman function based generalized proximal algorithms for each of these formulations generating three classes of complementarity algorithms the primal class is well known the dual class is new and constitutes a general collection of methods of multipliers or augmented lagrangian methods for complementarity problems in a special case it corresponds to a class of variational inequality algorithms proposed by gabay by appropriate choice of bregman function the augmented lagrangian subproblem in these methods can be made continuously differentiable the primal dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems

hyperplane arrangement cohomology and monomials in the exterior algebra we show that if x is the complement of a complex hyperplane arrangement then the homology of x has linear free resolution as a module over the exterior algebra on the first cohomology of x we study invariants of x that can be deduced from this resolution a key ingredient is a result of aramova avramov and herzog 2000 on resolutions of monomial ideals in the exterior algebra we give a new conceptual proof of this result

on cohomology algebras of complex subspace arrangements the integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement we give a combinatorial presentation of the cohomology algebra in the spirit of the orlik solomon result on the cohomology algebras of complex hyperplane arrangements our methods are elementary we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space we describe simplicial cochains that generate the cohomology among them we distinguish a linear basis study cup product multiplication and derive an algebra presentation in terms of generators and relations

equivariant cohomology and the varchenko gelfand filtration abstract the cohomology of the configuration space of n points in r 3 is isomorphic to the regular representation of the symmetric group which acts by permuting the points we give a new proof of this fact by showing that the cohomology ring is canonically isomorphic to the associated graded of the varchenko gelfand filtration on the cohomology of the configuration space of n points in r 1 along the way we give a presentation of the equivariant cohomology ring of the r 3 configuration space with respect to a circle acting on r 3 via rotation around a fixed line we extend our results to the settings of arbitrary real hyperplane arrangements the aforementioned theorems correspond to the braid arrangement as well as oriented matroids

group actions on arrangements of linear subspaces and applications to configuration spaces for an arrangement of linear subspaces in ri that is invariant under a finite subgroup of the general linear group gin ir we develop a formula for the g module structure of the cohomology of the complement m a our formula specializes to the well known goresky macpherson theorem in case g 1 but for g 1 the formula shows that the g module structure of the complement is not a combinatorial invariant as an application we are able to describe the free part of the cohomology of the quotient space ma4ig our motivating examples are arrangements in cn that are invariant under the action of sn by permuting coordinates a particular case is the k equal arrangement first studied by bj6rner lovasz and yao motivated by questions in complexity theory in these cases ma4 and ma sn are spaces of ordered and unordered point configurations in cn many of whose properties are reduced by our formulas to combinatorial questions in partition lattices more generally we treat point configurations in rd and provide explicit results for the k equal and the k divisible cases

perverse sheaves over real hyperplane arrangements let h be an arrangement of real hyperplanes in r the complexification of h defines a natural stratification of c we denote by perv c h the category of perverse sheaves on c smooth with respect to this stratification we give a description of perv c h as the category of representations of an explicit quiver with relations whose vertices correspond to real faces of h of all dimensions the relations are of monomial nature they identify some pairs of paths in the quiver they can be formulated in terms of the oriented matroid associated to h

topology and geometry of cohomology jump loci we elucidate the key role played by formality in the theory of characteristic and resonance varieties we define relative characteristic and resonance varieties v k and r k related to twisted group cohomology with coefficients of arbitrary rank we show that the germs at the origin of v k and r k are analytically isomorphic if the group is 1 formal in particular the tangent cone to v k at 1 equals r k these new obstructions to 1 formality lead to a striking rationality property of the usual resonance varieties a detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth complex quasi projective algebraic variety this sheds new light on a classical problem of j p serre applications to arrangements configuration spaces coproducts of groups and artin groups are given

from moment graphs to intersection cohomology we describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori in terms of structure of the zero and one dimensional orbits the class of varieties to which our formula applies includes schubert varieties in flag varieties and affine flag varieties we also prove a monotonicity result on local intersection cohomology stalks

perverse sheaves and actions where t x denotes the action of t on x the set xw is known to be a locally closed c stable algebraic subvariety of x isomorphic to an affine space the pieces xw form a cell decomposition x uwewxw w e w the socalled bialynicki birula decomposition we assume this decomposition to be an algebraic stratification of x the closure of a cell may not be a union of cells in general let zw be the intersection cohomology complex on the closure of a cell xw extended by 0 to the whole of x the hypercohomology h 2w has a natural structure of a graded h x c module further for any x y e w there is a natural morphism of graded spaces

combinatorial geometries convex polyhedra and schubert cells this paper is a continuation of gm which was published in the same journal we will explore a remarkable connection between the geometry of the schubert cells in the grassmann manifold the theory of convex polyhedra and the theory of combinatorial geometries in the sense of crapo and rota cr the results in this paper were obtained simultaneously and independently by gelfand and serganova in moscow and by goresky and macpherson at the i h e s in paris as part of larger programs with different purposes see below the geometry of this simple example is so beautiful that we decided to publish it independently of the applications we believe that combinatorial methods will play an increasing role in the future of geometry and topology we consider the grassmann manifold gi of all n k dimensional subspaces of c by fixing the standard basis in c we obtain an action of the torus h i on g pk which is induced from stretching the coordinate axes in c see also sect 1 we will describe not only the trajectories but also the strata of a new and interesting decomposition of the 301 oool 8708 87 7 50

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

the minimal model of the complement of an arrangement of hyperplanes in this paper the methods of rational homotopy theory are applied to a family of examples from singularity theory let a be a finite collection of hyperplanes in cl and let m cl uhea h we say a is a rational k w 1 arrangement if the rational completion of m is aspherical for these arrangements an identity the lcs formula is established relating the lower central series of 7r1 m to the cohomology of m this identity was established by group theoretic means for the class of fiber type arrangements in previous work we reproduce this result by showing that the class of rational k w 1 arrangements contains all fiber type arrangements this class includes the reflection arrangements of types al and bl there is much interest in arrangements for which m is a k w 1 space the methods developed here do not apply directly because m is rarely a nilpotent space we give examples of k w 1 arrangements which are not rational k w 1 for which the lcs formula fails and k w 1 arrangements which are not rational k w 1 where the lcs formula holds it remains an open question whether rational k w 1 arrangements are necessarily k w 1 l introduction an arrangement of hyperplanes is a finite collection of clinear subspaces of dimension 1 l in c1 to such an arrangement a is associated an open 21 manifold the complement m cl u hih e a the connections between the topology of m and the combinatorial geometry of a are the source of much current research in this area the most successful investigations concern the cohomology of m 13 15 whereas the most difficult unsolved problems involve the homotopy groups of m 5 in this paper we study the link between cohomology and homotopy provided by sullivan s theory of minimal models 14 in 6 a numerical relationship was established between 1 m and h m for the class of fiber type arrangements this lcs formula reads as follows ri 1 tn y7n m pm t n l where the n m are the ranks of successive quotients in the lower central series of 1 m and pm t is the poincare polynomial of h m because the sequence n m is related to the l minimal model y of m we conjecture in 5 that the lds formula holds precisely when r determines h m with the methods developed in this paper we can resolve this conjecture specifically we show corollary 3 8 that the lcs formula holds when h 5 is isomorphic to h m arrangements satisfying the latter condition are called rational k 7r l corollary 3 8 may be received by the editors june 1 1987 1980 mathematicts subject clatstsification 1985 revitsion primary 55p62 57m05 secondary 06c10 r 1988 american mathematical society 0002 9947 88 1 00 25 per page

on the algebra associated with a geometric lattice abstract let l be a geometric lattice following p orlik and l solomon combinatorics and topology of complements of hyperplanes invent math 56 1980 167 189 we associate with l a graded commutative algebra a l in this paper we introduce a new invariant of the algebra a l which suffices to distinguish algebras for which all other known invariants coincide this result is applied to the study of arrangements of complex hyperplanes with l being the intersection lattice in this case a l is isomorphic to the cohomology algebra of the associated hyperplane complement the goal is to find examples of arrangements with non isomorphic lattices but homotopy equivalent complements the invariant introduced here effectively narrows the list of candidates nevertheless we exhibit combinatorially inequivalent arrangements for which all known invariants including coincide

topological representations of matroids one of the foundations of oriented matroid theory is the topological representation theorem of folkman and lawrence 8 it says that an oriented simple matroid can be realized uniquely as an arrangement of pseudospheres that there is no similar interpretation for the class of all matroids has been taken for granted for instance a non coordinatizable matroid of abstract origin may be thought of as a geometric object only in a purely formal way whereas an oriented matroid may always be thought of as a geometric topological configuration on the d sphere or in projective space 3 p 19 our main theorem is that the class of geometric lattices which is cryptomorphic to the category of simple matroids is the same as the class of intersection lattices of arrangements of homotopy spheres the interpretation of a geometric lattice as an arrangement of homotopy spheres is a natural generalization of the folkman lawrence theorem an oriented matroid realizable over r has a representation with geodesic spheres allowing pseudospheres i e those which are homeomorphic but possibly not isometric to the unit sphere leads to the category of simple oriented matroids if we further relax the conditions on the spheres to only homotopy equivalence to the standard sphere then we are led to the category of all simple matroids some of the theory of oriented matroids which only depends on the underlying matroid can be extended to homotopy sphere arrangements zaslavsky s enumerative theory for pseudosphere arrangements can be extended to the homotopy setting as in the oriented matroid representation theorem the arrangement can be forced to be antipodal so a realization as homotopy projective spaces is also possible the minimal cellular resolutions of orientable matroid ideals developed in 10 can be extended to arbitrary matroids by using arrangements of homotopy spheres our point of view is primarily through the lens of oriented matroids hence the homotopy spheres which represent the atoms of the geometric lattice have codimension one in the future we hope to examine the point of view of complex hyperplane arrangements and consider spheres of even codimension the matroid theory we require is in section 2 while section 3 presents matroid steiner complexes we review a very general theory of arrangements of topological

linear convergence of epsilon subgradient descent methods for a class of convex functions this paper establishes a linear convergence rate for a class of epsilon subgradient descent methods for minimizing certain convex functions currently prominent methods belonging to this class include the resolvent proximal point method and the bundle method in proximal form considered as a sequence of serious steps other methods such as the recently proposed descent proximal level method may also fit this framework depending on implementation the convex functions covered by the analysis are those whose conjugates have subdifferentials that are locally upper lipschitzian at the origin a class introduced by zhang and treiman we argue that this class is a natural candidate for study in connection with minimization algorithms

homotopy types of subspace arrangements via diagrams of spaces we prove combinatorial formulas for the homotopy type of the union of the subspaces in an affine compactified affine spherical or projective subspace arrangement from these formulas we derive results of goresky macpherson on the homology of the arrangement and the cohomology of its complement the union of an arrangement can be interpreted as the direct limit of a diagram of spaces over the intersection poset a closely related space is obtained by taking the homotopy direct limit of this diagram our method consists in constructing a combinatorial model diagram over the same poset whose homotopy limit can be compared to the original one by usual homotopy comparison results for diagrams of spaces

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

inexact variants of the proximal point algorithm without monotonicity this paper studies convergence properties of inexact variants of the proximal point algorithm when applied to a certain class of nonmonotone mappings the presented algorithms allow for constant relative errors in the line of the recently proposed hybrid proximal extragradient algorithm the main convergence result extends a recent work of the second author where exact solutions for the proximal subproblems were required we also show that the linear convergence property is preserved in the case when the inverse of the operator is locally lipschitz continuous near the origin as an application we give a convergence analysis for an inexact version of the proximal method of multipliers for a rather general family of problems which includes variational inequalities and constrained optimization problems

k parabolic subspace arrangements in this paper we study k parabolic arrangements a generalization of the k equal arrangement for any finite real reflection group when k 2 these arrangements correspond to the well studied coxeter arrangements brieskorn 1971 showed that the fundamental group of the complement of the type w coxeter arrangement over mathbb c is isomorphic to the pure artin group of type w khovanov 1996 gave an algebraic description for the fundamental group of the complement of the 3 equal arrangement over mathbb r we generalize khovanov s result to obtain an algebraic description of the fundamental group of the complement of the 3 parabolic arrangement for arbitrary finite reflection group our description is a real analogue to brieskorn s description

on the homotopy type of the quillen complex of finite soluble groups abstract we prove that the homotopy type of the quillen complex of a finite soluble group at the prime p 2 is that of a wedge of spheres of possible different dimensions

geometric langlands duality and representations of algebraic groups over commutative rings as such it can be viewed as a first step in the geometric langlands program the connected complex reductive groups have a combinatorial classification by their root data in the root datum the roots and the co roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs if g is a reductive group we write g for its companion and call it the dual group g the notion of the dual group itself does not appear in satake s paper but was introduced by langlands together with its various elaborations in li l2 and is a cornerstone of the langlands program it also appeared later in physics mo gno in this paper we discuss the basic relationship between g and g we begin with a reductive g and consider the affine grassmannian qx the grassmannian for the loop group of g for technical reasons we work with formal algebraic loops the affine grassmannian is an infinite dimen sional complex space we consider a certain category of sheaves the spherical perverse sheaves on r these sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a hopf algebra by what has come to be known as tannakian formalism the resulting hopf algebra turns out to be the ring of functions on g in this interpretation the spherical perverse sheaves on the affine grassman nian correspond to finite dimensional complex representations of g thus instead of defining g in terms of the classification of reductive groups we pro vide a canonical construction of g starting from g we can carry out our construction over the integers the spherical perverse sheaves are then those with integral coefficients but the grassmannian remains a complex algebraic object

decompositions of the polyhedral product functor with applications to moment angle complexes and related spaces this article gives a natural decomposition of the suspension of a generalized moment angle complex or partial product space which arises as the polyhedral product functor described below the introduction and application of the smash product moment angle complex provides a precise identification of the stable homotopy type of the values of the polyhedral product functor one direct consequence is an analysis of the associated cohomology for the special case of the complements of certain subspace arrangements the geometrical decomposition implies the homological decomposition in earlier work of others as described below because the splitting is geometric an analogous homological decomposition for a generalized moment angle complex applies for any homology theory implied therefore is a decomposition for the stanley reisner ring of a finite simplicial complex and natural generalizations

real quadrics in cn complex manifolds and convex polytopes in this paper we investigate the topology of a class of non kahler compact complex manifolds generalizing that of hopf and calabi eckmann manifolds these manifolds are diffeomorphic to special systems of real quadrics cn which are invariant with respect to the natural action of the real torus s1 n onto cn the quotient space is a simple convex polytope the problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes we prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich we also resolve an associated wall crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope finally as a nice consequence we obtain that affine non kahler compact complex manifolds can have arbitrary amount of torsion in their homology groups contrasting with the kahler situation

the cohomology rings of complements of subspace arrangements the ring structure of the integral cohomology of complements of real linear subspace arrangements is considered while the additive structure of the cohomology is given in terms of the intersection poset and dimension function by a theorem of goresky and macpherson we describe the multiplicative structure in terms of the intersection poset the dimension function and orientations of the participating subspaces for the class of arrangements without pairs of intersections of codimension one in particular this yields a description of the integral cohomology ring of complex arrangements conjectured by yuzvinsky for general real arrangements a weaker result is obtained the approach is geometric and the methods are elementary

the hypertoric intersection cohomology ring we present a functorial computation of the equivariant intersection cohomology of a hypertoric variety and endow it with a natural ring structure when the hyperplane arrangement associated with the hypertoric variety is unimodular we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category the computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset

higher homotopy groups of complements of complex hyperplane arrangements abstract we generalize results of hattori on the topology of complements of hyperplane arrangements from the class of generic arrangements to the much broader class of hypersolvable arrangements we show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range and we give an explicit z 1 module presentation of p the first non vanishing higher homotopy group we also give a combinatorial formula for the 1 coinvariants of p for affine line arrangements whose cones are hypersolvable we provide a minimal resolution of 2 and study some of the properties of this module for graphic arrangements associated to graphs with no 3 cycles the algorithm for computing 2 is purely combinatorial the fitting varieties associated to 2 may distinguish the homotopy 2 types of arrangement complements with the same 1 and the same betti numbers in low degrees

alternating projection proximal methods for convex programming and variational inequalities we consider a mixed problem composed in part of finding a zero of a maximal monotone operator and in part of solving a monotone variational inequality problem we propose a solution method for this problem that alternates between a proximal step for the maximal monotone operator part and a projection type step for the monotone variational inequality part and analyze its convergence and rate of convergence this method extends a decomposition method of chen and teboulle math programming 64 1994 pp 81 101 for convex programming and yields as a by product new decomposition methods

computational topology of equivariant maps from spheres to complements of arrangements the problem of the existence of an equivariant map is a classical topological problem ubiquitous in topology and its applications many problems in discrete geometry and combinatorics have been reduced to such a question and many of them resolved by the use of equivariant obstruction theory a variety of concrete techniques for evaluating equivariant obstruction classes are introduced discussed and illustrated by explicit calculations the emphasis is on d 2n equivariant maps from spheres to complements of arrangements motivated by the problem of finding a 4 fan partition of 2 spherical measures where d 2n is the dihedral group one of the technical highlights is the determination of the d 2n module structure of the homology of the complement of the appropriate subspace arrangement based on the geometric interpretation for the generators of the homology groups of arrangements

poset fiber theorems suppose that f p q is a poset map whose fibers f 1 qless than or equal to q are sufficiently well connected our main result is a formula expressing the homotopy type of p in terms of q and the fibers several fiber theorems from the literature due to babson baclawski and quillen are obtained as consequences or special cases homology cohen macaulay and equivariant versions are given and some applications are discussed

composition duality and maximal monotonicity this paper shows how the attouch thera duality principle for operator inclusions can be extended to compositions of multifunctions so that the primal and dual inclusions may involve operators between different pairs of spaces we first present the extension and give an example from computational economics to demonstrate that it has practical utility then we consider a particular case often found in applications involving real hilbert spaces and maximal monotone operators we show that in this case our duality framework includes those previously developed by rockafellar mosco and gabay finally we demonstrate that in this case under very simple hypotheses the duality transformation preserves the maximal monotonicity of the operators involved this proof uses an apparently new criterion for maximal monotonicity of operators of the form l tl where t is maximal monotone and l is linear and continuous with adjoint l

hard lefschetz theorem for nonrational polytopes the hard lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope one can construct the intersection cohomology combinatorially from the polytope hence it is well defined even for nonrational polytopes when there is no variety associated to it we prove the hard lefschetz theorem for the intersection cohomology of a general polytope

response statistics of strongly nonlinear system to random narrowband excitation abstract a technique coupling with the parameter transformation method and the multiple scales method is presented for determining the primary resonance response of strongly nonlinear duffing rayleigh oscillator subject to random narrowband excitation by introducing a new expansion parameter e u 0 the multiple scales method is adapted to determine the equations describing the modulation of response amplitude and phase the effect of the random excitation on the stable periodic response is analyzed as a perturbation by the moment method steady state mean square response is obtained and its local stability is checked by routh hurwitz criterion theoretical analyses and numerical calculations show that when the intensity of random excitation increases the steady state solution may change from a limit cycle to a diffused limit cycle under some conditions the system may have two steady state solutions the results obtained for strongly nonlinear oscillator complement previous results in the literature for weakly nonlinear case

finite termination of the proximal point algorithm this paper concerns the notion of a sharp minimum on a set and its relationship to the proximal point algorithm we give several equivalent definitions of the property and use the notion to prove finite termination of the proximal point algorithm