on the computational complexity of piecewise linear homotopy algorithms we show that piecewise linear homotopy algorithms may take a number of steps that grows exponentially with the dimension when solving a system of linear equations whose solution lies close to the starting point our examples are based on an example of murty exhibiting exponential growth for lemke s algorithm for the linear complementarity problem

approximate labelling for simplicial algorithms and two classes of special subsets of the sphere call a subset of rn complete if the origin is in its convex hull we are concerned with the construction of large finite sets of points on the unit sphere whose complete subsets are in some sense easy to describe our motivation comes from simplicial algorithms for approximating zeroes of functions we construct two classes of such subsets of the sphere both of which generalize a subset that corresponds to usual integer labelling

solving discrete zero point problems in this paper we present two theorems on the existence of a discrete zero point of a function from the n dimensional integer lattice n to the n dimensional euclidean space n the theorems differ in their boundary conditions for both theorems we give a proof using a combinatorial lemma and present a constructive proof based on a simplicial algorithm that finds a discrete zero point within a finite number of steps

the buzzard diamond jarvis conjecture for unitary groups let p 2 be prime we prove the weight part of serre s conjecture for rank two unitary groups for mod p representations in the unramified case that is the buzzard diamond jarvis conjecture for unitary groups by proving that any serre weight which occurs is a predicted weight our methods are purely local using the theory of modules to determine the possible reductions of certain two dimensional crystalline representations

an efficient simplicial algorithm for computing a zero of a convex union of smooth functions we present an efficient simplicial algorithm for computing a zero of a point to set mapping that is formed by piecing together smooth functions such mappings arise in nonlinear programming and economic equilibrium problems our algorithm under suitable regularity conditions on the problem generates a sequence converging at least q superlinearly to a zero of the mapping asymptotically it operates in a space of reduced dimension analogous to an active set strategy in the optimization setting but it switches active sets automatically results of computational experiments are given

a simplicial approach to the determination of an integer point of a simplex we propose an algorithm to find an integer point of a simplex the algorithm is based on an integer labeling rule and a triangulation of the space starting at an integer point the algorithm leaves it along one of n rays and generates a sequence of adjacent simplices of varying dimension within a finite number of iterations the algorithm either yields an integer point of the simplex or proves that no such point exists

solving discrete systems of nonlinear equations abstract we study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice z n of the n dimensional euclidean space r n it is assumed that the set is integrally convex which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of z n and each simplex of the triangulation lies in an n dimensional cube of size one with respect to this triangulation we assume that the function satisfies some property that replaces continuity under this property and some boundary condition the function has a zero point to prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations the standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied because the continuity property is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself we apply the main existence result to prove the existence of a pure cournot nash equilibrium in a cournot oligopoly model we further obtain a discrete analogue of the well known borsuk ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem

variable dimension algorithms basic theory interpretations and extensions of some existing methods in this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by van der laan and talman we introduce a new concept primal dual pair of subdivided manifolds and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms our basic model furnishes interpretations to several existing methods lemke s algorithm for the linear complementarity problem its extension to the nonlinear complementarity problem a variable dimension algorithm on conical subdivisions and merrill s algorithm we shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us a method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed

a variable dimension fixed point algorithm and the orientation of simplices a variable dimension algorithm with integer labelling is proposed for solving systems ofn equations inn variables the algorithm is an integer labelling version of the 2 ray algorithm proposed by the author the orientation of lower dimensional simplices is studied and is shown to be preserved along a sequence of adjacent simplices

weakly closed unipotent subgroups in chevalley groups abstract the goal of this note is to classify the weakly closed unipotent subgroups in the split chevalley groups in an application we show under some mild assumptions on the characteristic that the lie algebra of a connected simple algebraic group fails to be a so called 2f module

construction of central elements in the affine hecke algebra via nearby cycles 0 1 overview let g be a connected reductive group over a finite field fq and let g k be the corresponding group over the local field k fq t let g o g k be a maximal compact subgroup of g k here o fq t and let hsph denote the hecke algebra of g k with respect to g o in other words hsph as a vector space consists of compactly supported bi g o invariant functions g k q and the product is defined by

discrete fixed points models complexities and applications we study three discrete fixed point concept sperner dpzp brouwer under two different models the polynomial time function model and the oracle function model we fully characterize the computational complexities of these three problems the computational complexity unification of the above problems gives us more choices in the study of different applications as an example by a reduction from dpzp we derive asymptotically equal lower and upper bound for tucker in the oracle model the same reduction also allows us to derive a single proof for the ppad completeness of tucker in any constant dimension which is significantly simpler than the recent proofs

on the flatness of local models for the symplectic group abstract we investigate the bad reduction of certain shimura varieties associated to the symplectic group more precisely we look at a model of the shimura variety at a prime p with parahoric level structure at p we show that this model is flat as conjectured by rapoport and zink ann of math stud 141 1996 and that its special fibre is reduced a crucial ingredient is faltings theorem on the normality of schubert varieties in the affine flag variety

weight monodromy conjecture over equal characteristic local fields the aim of this paper is to study certain properties of the weight spectral sequences of rapoport zink by a specialization argument by reducing to the case over finite fields previously treated by deligne we prove that the weight filtration and the monodromy filtration defined on the l adic etale cohomology coincide up to shift for proper smooth varieties over equal characteristic local fields we also prove that the weight spectral sequences degenerate at e 2 in any characteristic without using log geometry moreover as an application we give a modulo p 0 reduction proof of a hodge analogue previously considered by steenbrink

on the flatness of models of certain shimura varieties of pel type consider a pel shimura variety associated to a unitary group that splits over an unramified extension of q p rapoport and zink have defined a model of the shimura variety over the ring of integers of the completion of the reflex field at a place lying over p with parahoric level structures at p we show that this model is flat as conjectured by rapoport and zink and that its special fibre is reduced

computing an integer point of a simplex with an arbitrary starting homotopy like simplicial algorithm abstract an arbitrary starting homotopy like simplicial algorithm is developed for computing an integer point of an n dimensional simplex the algorithm is derived from the use of an integer labeling rule and a triangulation of r n 0 1 and consists of two interchanging phases one phase of the algorithm constitutes a homotopy simplicial algorithm which generates n 1 dimensional simplices in r n 0 1 and the other phase of the algorithm constitutes a pivoting procedure which generates n dimensional simplices in either r n 0 or r n 1 the algorithm varies from one phase to the other when the matrix defining the simplex is in the so called canonical form starting at an arbitrary integer point in r n 0 the algorithm within a finite number of iterations either yields an integer point of the simplex or proves that no such point exists

overgroup lattices in finite groups of lie type containing a parabolic the main theorem is a step in a program to show there exist finite lattices that are not an interval in the lattice of subgroups of any finite group as part of the proof of the main theorem we prove a theorem on the structure of maximal parabolics in finite groups of lie type which is of independent interest

quadratic chabauty for modular curves and modular forms of rank one in this paper we provide refined sufficient conditions for the quadratic chabauty method on a curve x to produce an effective finite set of points containing the rational points x mathbb q with the condition on the rank of the jacobian of x replaced by condition on the rank of a quotient of the jacobian plus an associated space of chow heegner points we then apply this condition to prove the effective finiteness of x mathbb q for any modular curve x x 0 n or x mathrm ns n of genus at least 2 with n prime the proof relies on the existence of a quotient of their jacobians whose mordell weil rank is equal to its dimension and at least 2 which is proven via analytic estimates for orders of vanishing of l functions of modular forms thanks to a kolyvagin logachev type result

computational complexity of a piecewise linear homotopy algorithm we give a bound on the number of steps required by the piecewise linear algorithm based on component wise homotopy proposed by the author for structured problems when solving a linear problem when the coefficient matrix is symmetric and positive definite this bound is polynomial inn and linear in the condition number of the matrix we also investigate the expected value of the bound for a particular distribution of such matrices

uber die lokale zetafunktion von shimuravarietaten monodromiefiltration und verschwindende zyklen in ungleicher charakteristik

cells in affine weyl groups and tensor categories abstract in this paper we construct a tensor or monoidal category for any two sided cell in a finite or affine weyl group the isomorphism classes of simple objects of this category correspond to the elements of the two sided cell the tensor product arises by a certain truncation procedure from the standard convolution of perverse sheaves on a flag manifold which imitates geometrically the definition of the ring j given in g lusztig j algebra 109 1987 536 548 similarly we construct for any left cell a tensor category in which the isomorphism classes of simple objects correspond to the elements in the left cell intersected with its inverse a right cell this contains as a special case the category of equivariant perverse sheaves on an affine grassmannian which as a consequence of g lusztig asterisque 101 102 1983 208 209 is a tensor category for convolution without truncation closely related to the tensor category of representations of the group dual in the sense of langlands to the group which gives rise to the affine grassmannian but our construction can be specialized in other ways and it gives a conjectural realization of the tensor category of representations of the reductive quotient of the centralizer of any unipotent element of the langlands dual group

a new simplicial variable dimension algorithm to find equilibria on the product space of unit simplices a new simplicial variable dimension restart algorithm is introduced to solve the nonlinear complementarity problem on the product spaces of unit simplices the triangulation which underlies the algorithm differs from the triangulations ofs used thus far moreover the number of rays along which the algorithm can leave the arbitrarily chosen starting point is much larger more precisely there is a ray leading from the starting point to each vertex ofs in cases is the product ofn one dimensional unit simplices the alogrithm is similar to the octahedral algorithm onrn having 2n rays also the accuracy of an approximate solution in the terminal simplex of the algorithm is in general better than for the other algorithms ons computational results will show that the number of iterations for the new algorithm is much less the examples concern the computation of equilibria in noncooperative games exchange economies and trade models

a vector labeling method for solving discrete zero point and complementarity problems in this paper we establish the existence of a discrete zero point of a function from the n dimensional integer lattice mathbb z n to the n dimensional euclidean space mathbb r n under very general conditions with respect to the behavior of the function the proof is constructive and uses a combinatorial argument based on a simplicial algorithm with vector labeling and lexicographic linear programming pivot steps the algorithm provides an efficient method to find an exact solution we also discuss how to adapt the algorithm for two related problems namely to find a discrete zero point of a function under a general antipodal condition and to find a solution to a discrete nonlinear complementarity problem in both cases the modified algorithm provides a constructive existence proof too we further show that the algorithm for the discrete nonlinear complementarity problem generalizes the well known lemke s method to nonlinear environments an economic application is also presented

subgroups of lie type groups containing a unipotent radical abstract in this paper we show that if g is a lie type group and r a proper subgroup of g containing some unipotent radical then r has a nilpotent normal subgroup generated by long root subgroups and r is contained in a proper parabolic subgroup of g we also obtain some consequences of this result

complete reducibility of quadratic modules for finite lie type groups let g be a finite lie type group in char p 3 different from sl2 3 and v a zpg module on which the root groups of g act quadratically i e v ar ar 0 for all roots of the root system of g then we show that v is completely reducible

the octahedral algorithm a new simplicial fixed point algorithm a new variable dimension simplicial algorithm for the computation of solutions of systems of nonlinear equations or the computation of fixed points is presented it uses the restrart technique of merrill to improve the accuracy of the solution the algorithm is shown to converge quadratically under certain conditions the algorithm should be efficient and relatively easy to implement

formulae relating the bernstein and iwahori matsumoto presentations of an affine hecke algebra abstract we consider the anti dominant variants of the elements occurring in the bernstein presentation of an affine hecke algebra h we find explicit formulae for in terms of the iwahori matsumoto generators tw w ranging over the extended affine weyl group of the root system r in the case i r is arbitrary and is a minuscule coweight or ii r is attached to gln and mek where ek is a standard basis vector and m 1 in the above cases certain minimal expressions for play a crucial role such minimal expressions exist in fact for any coweight for gln we give a sheaf theoretic interpretation of the existence of a minimal expression for the corresponding perverse sheaf on the affine schubert variety x t is the push forward of an explicit perverse sheaf on the demazure resolution m x t x t this approach yields for a minuscule coweight of any r or for an arbitrary coweight of gln a conceptual albeit less explicit expression for the coefficient w of the basis element tw in terms of the cohomology of a fiber of the demazure resolution

a discrete multivariate mean value theorem with applications we establish a discrete multivariate mean value theorem for the class of positive maximum component sign preserving functions a constructive and combinatorial proof is given based upon a simplicial algorithm and vector labeling moreover we apply this theorem to a discrete nonlinear complementarity problem and an economic equilibrium problem with indivisibilities and show the existence of solutions in both problems under certain mild conditions

on the computation of fixed points in the product space of unit simplices and an application to noncooperative n person games in this paper an algorithm based on the principle of simplicial approximation is introduced to compute fixed points of upper semicontinuous point to set mappings from the product space s of unit simplices into itself the algorithm is a modification of an algorithm introduced in an earlier paper the main feature is that it starts with an arbitrary chosen point in s and that the triangulation of s depends on the starting point moreover the algorithm can terminate with a non full dimensional subsimplex yielding a good approximation an application is given for non cooperative n person games where s is the strategy space some computational experiences are given

on the solutions of discrete nonlinear complementarity and related problems this paper concerns the existence of solutions to the discrete nonlinear complementarity problem the problem is that of finding an integer vector x in the n dimensional euclidean space such that both x and f x are nonnegative and the inner product of x and f x is equal to zero where f is a nonlinear function mapping from the n dimensional euclidean space into itself several sufficient conditions are introduced to show the existence of solutions or a unique solution to the problem meanwhile we study the closely related discrete fixed point problem and provide sufficient conditions for the existence of discrete fixed points in addition two economic applications are discussed

a unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm we present a unified description of a class of restart fixed point algorithms including merrill s method and several variable dimension algorithms for their implementation on computers based on this description we show that some pivot saving techniques originally developed for the homotopy methods can be applied to the class we also propose a new variable dimension algorithm having 3 n 1 rays along which we can move toward a solution some numerical comparisons of the simplicial restart algorithms merrill s method the 2n method the octahedral method and the new one support that the latter two methods are more efficient than the others

the jordan holder series for nearby cycles on some shimura varieties and affine flag varieties we study the jordan holder series for nearby cycles on certain shimura varieties and rapoport zink local models and on finite dimensional pieces of beilinson s deformation of the affine grassmannian to the affine flag variety and their p adic analogues we give a formula for the multiplicities of irreducible constituents in terms of certain cohomology groups and we also provide an algorithm to compute multiplicities in terms of the affine hecke algebra msc classification 14m15 14g35 20c08

groups of lie type generated by long root elements in f4 k abstract let k be a field and g a quasi simple subgroup of the chevalley group f4 k we assume that g is generated by a class of abstract root subgroups such that there are a c with a c and any a is contained in a long root subgroup of f4 k we determine the possibilities for g and describe the embedding of g in f4 k

a refined beilinson bloch conjecture for motives of modular forms we propose a refined version of the beilinson bloch conjecture for the motive associated with a modular form of even weight this conjecture relates the dimension of the image of the relevant p adic abel jacobi map to certain combinations of heegner cycles on kuga sato varieties we prove theorems in the direction of the conjecture and in doing so obtain higher weight analogues of results for elliptic curves due to darmon

efficiency and implementation of simplicial zero point algorithms in this paper we compare the efficiency of several simplicial variable dimension algorithms to do so we first treat the issues of degeneracy and accelerating we present a device for solving degeneracy furthermore we compare several accelerating techniques the technique of iterated quasi newton steps after each major cycle of the simplicial algorithm is implemented in a computer code which is used to compare the efficiency of the n 1 ray 2n ray 2n ray and 3n 1 ray algorithms except for the n 1 ray algorithm the number of function evaluations does not differ very much between the various algorithms it appeared however that the 2n algorithm needs considerably less computation time

simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds ideas of a simplicial variable dimension restart algorithm to approximate zero points onrn developed by the authors and of a linear complementarity problem pivoting algorithm are combined to an algorithm for solving the nonlinear complementarity problem with lower and upper bounds the algorithm can be considered as a modification of the2n ray zero point finding algorithm onrn it appears that for the new algorithm the number of linear programming pivot steps is typically less than for the2n ray algorithm applied to an equivalent zero point problem this is caused by the fact that the algorithm utilizes the complementarity conditions on the variables

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 the existence and approximation of zeroes in this paper we consider the problem of finding zeroes of a continuous functionf from a convex compact subsetu of n to n in the first part of the paper it is proved thatf has a computable zero iff cn n satisfies the nonparallel condition for any two antipodal points on bdcn i e if for anyx bdcn f x f x 0 holds therefore we describe a simplicial algorithm to approximate such a zero it is shown that generally the degree of the approximate zero depends on the number of reflection steps made by the algorithm i e the number of times the algorithm switches from a face on bdcn to the face therefore the index of a terminal simplex is defined which equals the local brouwer degree of the function if is full dimensional in the second part of the paper the algorithm is used to generate possibly several approximate zeroes off two sucessive solutions may have both the same or opposite degrees again depending on the number of reflection steps by extendingf u n to a function g from a cube containingu to n the procedure can be applied to any continuous functionf without having any information about the global and local brouwer degrees a priori

alcoves associated to special fibers of local models the special fiber of the local model of a pel shimura variety with iwahori type level structure admits a cellular decomposition the set of strata is in a natural way a finite subset of the affine weyl group determined by the shimura data in this paper we generalize and give a new proof of a theorem of kottwitz and rapoport concerning the description of this set for the case of linear and symplectic groups for higher rank groups not of type an we produce counterexamples showing the situation is not as nice in general we also recover a theorem of deodhar characterizing the bruhat order on sn the local geometry can be approached using the rapoport zink local model m j 13 this protective op scheme is a local model for the singularities in the special fiber of sh g x9 k 9 but it is defined in terms of the pair g p using linear algebra and is somewhat easier to deal with than the shimura variety itself in this paper we address certain combinatorial questions which arise in the study of the special fiber m f of m to fix ideas consider the case where g is split it turns out that m f can be regarded as a finite dimensional union of iwahori orbits in the affine flag variety for g fp t see 5 and 8 it follows that m f has a stratification indexed by a finite set perm p of the extended affine weyl group w g for g equivalently perm p can be regarded as a finite set of alcoves in the affine coxeter complex determined by a splitting for g let wo denote the finite weyl group of g for each translation in the wo orbit

kloosterman sheaves for reductive groups deligne constructed a remarkable local system on p 1 f 0 1g attached to a family of kloosterman sums katz calculated its monodromy and asked whether there are kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the langlands correspondence motivated by work of gross and frenkel gross we nd an explicit family of such automorphic forms and even a simple family of automorphic sheaves in the framework of the geometric langlands program we use these automorphic sheaves to construct adic kloosterman sheaves for any reductive group in a uniform way and describe the local and global monodromy of these kloosterman sheaves in particular they give motivic galois representations with exceptional monodromy groups g2 f4 e7 and e8 this also gives an example of the geometric langlands correspondence with wild ramication for any reductive group

frobenius semisimplicity for convolution morphisms this article concerns properties of mixed ell adic complexes on varieties over finite fields related to the action of the frobenius automorphism we establish a fiberwise criterion for the semisimplicity and frobenius semisimplicity of the direct image complex under a proper morphism of varieties over a finite field we conjecture that the direct image of the intersection complex on the domain is always semisimple and frobenius semisimple this conjecture would imply that a strong form of the decomposition theorem of beilinson bernstein deligne gabber is valid over finite fields we prove our conjecture for generalized convolution morphisms associated with partial affine flag varieties for split connected reductive groups over finite fields as a crucial tool we develop a new schematic theory of big cells for loop groups with suitable reformulations the main results are valid over any algebraically closed ground field

a new variable dimension algorithm for the fixed point problem we propose a new variable dimension algorithm for solving a system ofn equations inn variables the algorithm solves a sequence of subproblems consisting of the first several equations the dimension of subproblems does not however increase monotonically a convergence condition is also presented

local models of shimura varieties and a conjecture of kottwitz we give a group theoretic definition of local models as sought after in the theory of shimura varieties these are projective schemes over the integers of a p adic local field that are expected to model the singularities of integral models of shimura varieties with parahoric level structure our local models are certain mixed characteristic degenerations of grassmannian varieties they are obtained by extending constructions of beilinson drinfeld gaitsgory and the second named author to mixed characteristics and to the case of general tamely ramified reductive groups we study the singularities of local models and hence also of the corresponding integral models of shimura varieties in particular we study the monodromy inertia action and show a commutativity property for the sheaves of nearby cycles as a result we prove a conjecture of kottwitz which asserts that the semi simple trace of frobenius on the nearby cycles gives a function which is central in the parahoric hecke algebra

a lexicographic algebraic theorem and its applications abstract in this paper we prove the following result given any full dimensional simple polytope p x r n a i t x b i i 1 m without redundant constraints and any vector c r n there exists a unique vertex x of p such that the matrix 1 1 1 1 c a i 1 a i 2 a i n 1 exists and is lexicographic positive where x is the solution of the n linear equations a i t t x b i t t 1 n this theorem generalizes a result on the n dimensional cube and can be used to resolve the degeneracy problem for simplicial fixed point algorithms we also discuss several applications for this result

a unifying model based on retraction for fixed point algorithms we present a unifying model based on retraction for several restart fixed point algorithms the model embraces the interpretation of the algorithms in terms of stationary point problem by van der laan and talman and fully explains the 2 ray method

a homotopy for solving large sparse and structured fixed point problems we consider here the problem of solving a system of n nonlinear equations in n variables when n is large but the underlying mapping has a sparse jacobian and is also structured we present a homotopy having a variable dimension feature whose implementation in a pl algorithm effectively exploits the sparsity of the jacobian and separability of the mapping the implementation given here uses the cholesky factorization and is thus stable an application to a large system is also discussed

a better triangulation for wright s 2n ray algorithm abstract we give a triangulation for wright s 2n ray algorithm to compute solutions of nonlinear equations according to measures of efficiency of triangulations it is better than any other available triangulation for the 2n ray algorithm

a class of simplicial restart fixed point algorithms without an extra dimension in an earlier paper we introduced an algorithm for approximating a fixed point of a mapping on the product space of unit simplices ideas of that paper are used to construct a class of triangulations ofrn more precisely for somek 1 k n and positive integersm1 mk with sumn a triangulation ofrn is obtained by triangulating the cells which are formed by taking the product of given triangulations ofrmj j 1 k the triangulation of each cell will be defined in relation to an arbitrarily chosen pointv inrn being the starting point of the algorithm fork n we obtain thek triangulation originally due to todd each element of the class can be used to find a simplex which approximates a fixed point of a mapping onrn by generating a unique path of adjacent simplices of variable dimension starting with the pointv

a modified integer labeling for complementarity algorithms nonlinear complementary problems can be solved by algorithms based on the principle of simplicial approximation we present such an algorithm that allows the choice of an arbitrary starting point without using an artificial labeling or an extra dimension furthermore the relation between the grid size of the triangulation and the accuracy of an approximate solution is investigated and it is shown that a combination of the algorithm with newton steps yields better accuracy finally the algorithm is adapted for solving systems of equations

computing the alternating trace of frobenius on the sheaves of nearby cycles on local models for gl4 and gl5 abstract consider a shimura variety defined over some number field and assume we have a model over the ring of integers at some prime of bad reduction it is then interesting to know the alternating trace of frobenius on the invariants under the inertia group of the sheaf of nearby cycles since these traces are related to the local factor of the hasse weil zeta function in this article we compute the semi simple alternating trace of frobenius for shimura varieties associated to the groups gu 2 2 and gu 3 2 with level structure of iwahori type by investigating the equations of the local model defined by rapoport and zink and by performing explicit blowing ups