note multiplicative interval variation of objective function coefficients in linear programming traditional sensitivity analysis of linear programming objective function coefficients concerns itself with variations in single parameters in a recent book gal developed a theoretical framework to determine the effect of multiple variations in the parameters in this paper we develop a methodology to implement and extend the theoretical results in the earlier work for interval variations of objective function coefficients this methodology uses necessary and sufficient conditions to determine all optimal solutions to the linear programming problem for objective function coefficients within given intervals the resulting procedure has been coded for computer use and tested on various types of test problems computational results and an example are presented

on support varieties and the humphreys conjecture in type a abstract let g be a reductive algebraic group scheme defined over f p and let g 1 denote the frobenius kernel of g to each finite dimensional g module m one can define the support variety v g 1 m which can be regarded as a g stable closed subvariety of the nilpotent cone a g module is called a tilting module if it has both good and weyl filtrations in 1997 it was conjectured by j e humphreys that when p h the support varieties of the indecomposable tilting modules align with the nilpotent orbits given by the lusztig bijection in this paper we shall verify this conjecture when g s l n 1 and p n 1

maximal inner boxes in parametric ae solution sets with linear shape we consider linear systems of equations a p x b p where the parameters p are linearly dependent and come from prescribed boxes and the sets of solutions defined in various ways which have linear boundary one fundamental problem is to compute a box being inside a parametric solution set we first consider parametric tolerable solution sets being convex polyhedrons for such solution sets we prove that finding a maximal inner box is an np hard problem this justifies our exponential linear programming methods for computing maximal inner boxes we also propose a polynomial heuristic that yields a large but not necessarily the maximal inner box next we discuss how to apply the presented linear programming methods for finding large inner estimations of general parametric ae solution sets with linear shape numerical examples illustrate the properties of the methods and their application

cohomology of tilting modules over quantum groups and t structures on derived categories of coherent sheaves the paper is concerned with cohomology of the small quantum group at a root of unity and of its upper triangular subalgebra with coefficients in a tilting module it turns out to be related to irreducible objects in the heart of a certain t structure on the derived category of equivariant coherent sheaves on the springer resolution and to equivariant coherent ic sheaves on the nil cone the support of the cohomology is described in terms of cells in affine weyl groups the basis in the grothendieck group provided by the cohomology modules is shown to coincide with the kazhdan lusztig basis as predicted by j humphreys and v ostrik

perturbed matrix inversion with application to lp simplex method abstract computation of the inverse of a perturbed matrix a d 1 appears within various areas of research and applications of particular importance is the sparse simplex where a is the basis matrix in this paper a procedure for the computing of the inverse matrix a d 1 is presented and discussed where a r n n is a given non singular matrix a 1 is already calculated and a d is a perturbed matrix of a by d where d is sparse the non singularity requirement for d is removed

parabolic kazhdan lusztig polynomials and schubert varieties abstract we shall give a description of the intersection cohomology groups of the schubert varieties in partial flag manifolds over symmetrizable kac moody lie algebras in terms of parabolic kazhdan lusztig polynomials introduced by deodhar

algorithms for linear programming problems with interval objective function coefficients this paper presents three algorithms for solving linear programming problems in which some or all of the objective function coefficients are specified in terms of intervals which algorithm is applicable depends upon a the number of interval objective function coefficients b the number of nonzero objective function coefficients and c whether or not the feasible region is bounded the algorithms output all extreme points and unbounded edge directions that are multiparametrically optimal with respect to the ranges placed on the objective function coefficients the algorithms are most suitable to linear programs in which the objective function coefficients are deterministic but are likely to vary from time period to time period as for example in blending problems

tolerance sensitivity and optimality bounds in linear programming traditional sensitivity analysis in linear programming usually focuses on variations of one coefficient or term at a time the tolerance approach was proposed to provide a decision maker with an effective and easy to use method to summarize the effects of simultaneous and independent changes in selected parameters in particular for variations of the objective function coefficients the approach gives a maximum tolerance percentage within which selected coefficients may vary from their estimated values within a priori limits while still retaining the same optimal basic feasible solution although an optimal solution may cease being optimal for variations beyond the maximum tolerance percentage it may still be close to optimal herein we characterize the potential loss of optimality for variations beyond the maximum tolerance percentage as a maximum regret function we consider theoretical properties of this function and propose a method to compute a relevant portion of it

a finiteness theorem for zero cycles over p adic fields let r be a henselian discrete valuation ring let x be a regular projective flat scheme over spec r with generalized semistable reduction we prove a bijectivity theorem for etale cycle class maps of the chow group of 1 cycles on x as an application we prove a finiteness theorem for the chow group of 0 cycles on a projective smooth variety over a p adic field

support varieties for infinitesimal group schemes the representation theory of a connected smooth affine group scheme over a field k of characteristic p 0 is faithfully captured by that of its family of frobenius kernels such frobenius kernels are examples of infinitesimal group schemes affine group schemes g whose coordinate hopf algebra k g is a finite dimensional local k algebra this paper presents a study of the cohomology algebra h g k of an arbitrary infinitesimal group scheme over k we provide a geometric determination of the cohomological support variety cgl spec h g k analogous to that given by d quillen for the cohomology of finite groups q we further study finite dimensional rational g modules m for arbitrary infinitesimal group schemes g over k in a manner initiated by j alperin and l evens a e and j carlson ci for finite groups we consider the variety igim c igi of the ideal im ker hev g k ext m m and provide a geometric description of this variety which is analogous to that given by g avrunin and l scott for finite dimensional modules for finite groups a s this paper is a continuation of our recent work establishing the finite generation of h g k f s and investigating the infinitesimal 1 parameter subgroups of g s f b earlier work of e friedlander and b parshall fp1 fp2 fp3 fp4 and j jantzen ji concerning the cohomology of restricted lie algebras are forerunners of the results presented here finite dimensional restricted lie algebras are in 1 1 correspondence with infinitesimal group schemes of height 1 our main theorems theorems 5 2 and 6 7 below when restricted to infinitesimal group schemes of height 1 significantly strengthen previously known cohomological information for restricted lie algebras an interesting aspect of our work is the extent to which infinitesimal 1 parameter subgroups v g a r g for infinitesimal group schemes g of height r play the role of elementary abelian p subgroups and their generalizations shifted subgroups for finite groups indeed much of our effort is dedicated to proving that cohomology classes are detected modulo nilpotence by such 1 parameter subgroups this is first done in 2 for unipotent infinitesimal group schemes using an induction argument made possible by a structure theorem presented in 1 this structure theorem is the analogue in our context of a theorem of j p serre characterizing elementary abelian p groups s

some new examples in 1 cohomology this paper gives some new examples in the 1 cohomology theory of finite groups of lie type obtained from both computer calculations and the use of several theoretical results in particular the paper gives the first known examples of 1 cohomology groups of dimension greater than 2 for absolutely irreducible faithful modules of a finite group the computer calculations were made originally while checking special cases of lusztig s conjecture on characteristic p representations of algebraic groups and we take this opportunity to announce in print some results in that direction they reinforce lusztig s conjecture even in a strong form suggested by kato 2003 published by elsevier science usa

generalized linear fractional programming under interval uncertainty abstract data in many real life engineering and economical problems suffer from inexactness herein we assume that we are given some intervals in which the data can simultaneously and independently perturb we consider a generalized linear fractional programming problem with interval data and present an efficient method for computing the range of optimal values the method reduces the problem to solving from two to four real valued generalized linear fractional programs which can be computed in polynomial time using an appropriate interior point method solver we consider also the inverse problem how much can data of a real generalized linear fractional program vary such that the optimal values do not exceed some prescribed bounds we propose a method for calculating often the largest possible ranges of admissible variations it needs to solve only two real valued generalized linear fractional programs we illustrate the approach on a simple von neumann economic growth model

some topics on frobenius lusztig kernels i abstract an alternative description of the frobenius lusztig kernels and restricted specializations is presented and used to find their coradical filtrations automorphisms and imbeddings moreover factor hopf algebras of the frobenius lusztig kernels are determined

multi parametric analysis of the maximum tolerance in a linear programming problem abstract this study intends to propose an alternative approach to the multi parametric sensitivity analysis of a linear programming problem by the concept of a maximum volume in the tolerance region for the purposes of management and control the study first proposes the necessary and sufficient conditions of the focal parameters then those nonfocal parameters that have unlimited variations can be identified and thus because of their low sensitivity in practice and simplicity in analysis they will be deleted for those focal parameters their different levels of sensitivity are investigated and conceived by the proposed maximum volume region so that they can be handled with the greatest flexibility simultaneously and independently this maximum volume region is bounded by a symmetrically rectangular parallelepiped it can be characterized by a maximization problem and solved by the existing technique dynamic programming besides an extension to the extended maximum volume region for the case of semi bounded region is considered theoretical proofs are provided with numerical illustrations the result has been compared to wendell s approach

a fresh view on the tolerance approach to sensitivity analysis in linear programming abstract the tolerance approach to sensitivity analysis in linear programming aims at finding a unique numerical value tolerance representing the maximum absolute perturbation which can be applied simultaneously and independently on each right hand side or objective coefficient without affecting the optimality of the given basis some extensions have been proposed in the literature which allow for individual tolerances for each coefficient thus enlarging the tolerance region in this paper we review the main results concerning the approach giving new and simpler proofs and we propose an efficient geometric algorithm returning a tolerance region that is maximal with respect to inclusion we compare our method with the existing ones on two examples showing how a priori information can be naturally exploited by our algorithm to further enlarge individual tolerances

parabolic kazhdan lusztig r polynomials for quasi minuscule quotients abstract we give explicit combinatorial formulas for the parabolic kazhdan lusztig r polynomials of the quasi minuscule quotients of the classical weyl groups as an application of our results we obtain explicit combinatorial formulas for certain sums and alternating sums of ordinary kazhdan lusztig r polynomials

sensitivity analysis in linear programming just be careful in this paper we review the topic of sensitivity analysis in linear programming we describe the problems that may occur when using standard software and advocate a framework for performing complete sensitivity analysis three approaches can be incorporated within it one using bases an approach using the optimal partition and one using optimal values we elucidate problems and solutions with an academic example and give results from an implementation of these approaches to a large practical linear programming model of an oil refinery this shows that the approaches are viable and useful in practice

using bounds on the data in linear programming the tolerance approach to sensitivity analysis in contrast to traditional sensitivity analysis in linear programming the tolerance approach considers simultaneous and independent variations in a number of parameters a primary focus of this approach is to determine a maximum tolerance percentage for selected right hand side terms in which the same basis is optimal as long as each term is accurate to within that percentage of its estimated value similarly the approach yields a maximum tolerance percentage for selected objective function coefficients this paper shows how the tolerance approach can exploit information on the range of possible values over which terms and coefficients can vary to yield larger maximum tolerance percentages

differentiating the weyl generic dimension formula with applications to support varieties abstract the authors compute the support varieties of all irreducible modules for the small quantum group u g where g is a finite dimensional simple complex lie algebra and is a primitive l th root of unity with l larger than the coxeter number of g the calculation employs the prior calculations and techniques of ostrik and of nakano parshall and vella as well as deep results involving the validity of the lusztig character formula for quantum groups and the positivity of parabolic kazhdan lusztig polynomials for the affine weyl group analogous support variety calculations are provided for the first frobenius kernel g 1 of a reductive algebraic group scheme g defined over the prime field f p

support varieties of line bundle cohomology groups for sl3 k abstract let g sl 3 k where k is a field of characteristic p 0 and let x t be any weight with corresponding line bundle l on g b in this paper we compute the support varieties for all modules of the form h i h i g b l over the first frobenius kernel g 1 the calculation involves certain recursive character formulas given by donkin cf 5 which can be used to compute the characters of the line bundle cohomology groups in the case where is a p regular weight and m h i 0 for some i these formulas are used to show that any pth root of unity is not a root of the generic dimension of m to handle the case where is not p regular we employ techniques similar to those used by drupieski nakano and parshall cf 7 to show that the module h i is not projective over g 1 whenever it is nonzero and lies outside of the steinberg block

the tolerance approach to sensitivity analysis in linear programming in constrast to ordinary sensitivity analysis in linear programming the tolerance approach considers simultaneous and independent changes in the objective function coefficients and in the right hand side terms this approach yields a maximum tolerance percentage such that as long as selected coefficients or terms are accurate to within that percentage of their estimated values the same basis is optimal in particular if the objective function coefficients are accurate to within the maximum tolerance percentage of their specified values then the same solution is optimal

rationally connected varieties over local fields let x be a proper variety defined over a field k following ma we say that two points x x x k are r equivalent if they can be connected by a chain of rational curves defined over k cf 4 1 in essence two points are r equivalent if they are obviously rationally equivalent several authors have proved finiteness results over local and global fields cubic hypersurfaces ma sd linear algebraic groups ct sa vo1 gi vo2 rational surfaces ct co ct1 ct sk1 quadric bundles and intersections of two quadrics ct sa sd pa su r equivalence is only interesting if there are plenty of rational curves on x at least over k such varieties have been studied in the series of papers komi mo1 ko mi mo3 see also ko1 there are many a priori different ways of defining what plenty of rational curves should mean fortunately many of these turn out to be equivalent and this leads to the notion of rationally connected varieties see ko mi mo2 ko1 iv 3 ko2 4 1 2

infinitesimal 1 parameter subgroups and cohomology this is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p 0 whereas sfb is concerned with detection of cohomology classes the present paper introduces the graded algebra k vr g of functions on the scheme of infinitesimal 1 parameter subgroups of height r on an affine group scheme g and demonstrates that this algebra is essentially a retract of h g k provided that g is an infinitesimal group scheme of height kr this work is a continuation of f s in which the existence of certain universal extension classes was established thereby enabling the proof of finite generation of h g k for any finite group scheme g over k the role of the scheme of infinitesimal 1 parameter subgroups of g was foreshadowed in f p where h g 1 k was shown to be isomorphic to the coordinate algebra of the scheme of p nilpotent elements of g lie g for g a smooth reductive group g 1 the first frobenius kernel of g and p char k sufficiently large indeed p nilpotent elements of g correspond precisely to infinitesimal 1 parameter subgroups of g 1 much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1 parameter subgroups in 1 we construct the affine scheme vr g of homomorphisms from ga r to g which we call the scheme of infinitesimal 1 parameter subgroups of height r in g in the special case r 1 this is the scheme of p nilpotent elements of the p restricted lie algebra lie g for various classical groups g vr g is the scheme of r tuples of p nilpotent pairwise commuting elements of lie g more generally an embedding g c gln determines a closed embedding of vr g into the scheme of r tuples of p nilpotent pairwise commuting elements of gl1 lie gln the relationship between k vr g the coordinate algebra of vr g and h g k is introduced in theorem 1 14 a natural homomorphism of graded k algebras

surjectivity of the total clifford invariant and brauer dimension abstract a celebrated theorem of merkurjev that the 2 torsion of the brauer group is represented by clifford algebras of quadratic forms is in general false when the base is no longer a field the first counterexamples when the base is among certain arithmetically subtle hyperelliptic curves over local fields were constructed by parimala scharlau and sridharan we prove that considering clifford algebras of all line bundle valued quadratic forms such counterexamples disappear and we recover merkurjev s theorem in these cases for any smooth curve over a local field or any smooth surface over a finite field the 2 torsion of the brauer group is always represented by clifford algebras of line bundle valued quadratic forms

on chow and brauer groups of a product of mumford curves let c1 cd be mumford curves defined over a finite extension of open image in new window and let x c1 cd we shall show the following 1 the cycle map ch0 x n h2d x n d is injective for any non zero integer n 2 the kernel of the canonical map ch0 x hom br x open image in new window defined by the brauer manin pairing coincides with the maximal divisible subgroup in ch0 x

construction of the largest sensitivity region for general linear programs this paper develops an alternative approach to post optimality analysis for general linear program models which provides the largest sensitivity region of any single or simultaneous change of right hand side of constraints and the coefficients of decision variables in the objective function the goal is a theoretical unification of various types of sensitivity analyses as well as advancement in the practical implementation of post optimality analysis we extend our proposed approach in the construction of sensitivity region to maintain the degenerate vertex for models with degenerate optimal solution as well as maintaining the multiple solutions for the models with non unique optimal solutions as a by product the paper resolves the paradoxical situation known as the more for less less for more situations the proposed method is based on the given optimal solution s therefore it is easy to understand easy to implement and provides useful information to the decision makers the methodology and their computational algorithms are presented and discussed in the context of some illustrative numerical examples

torsion algebraic cycles and complex cobordism atiyah and hirzebruch gave the first counterexamples to the hodge conjecture with integral coefficients 3 that conjecture predicted that every integral cohomology class of hodge type p p on a smooth projective variety should be the class of an algebraic cycle but atiyah and hirzebruch found additional topological properties which must be satisfied by the integral cohomology class of an algebraic cycle here we provide a more systematic explanation for their results by showing that the classical cycle map from algebraic cycles modulo algebraic equivalence to integral cohomology factors naturally through a topologically defined ring which is richer than integral cohomology the new ring is based on complex cobordism a well developed topological theory which has been used only rarely in algebraic geometry since hirzebruch used it to prove the riemann roch theorem 17 this factorization of the classical cycle map implies the topological restrictions on algebraic cycles found by atiyah and hirzebruch it goes beyond their work by giving a topological method to show that the classical cycle map can be noninjective as well as nonsurjective the kernel of the classical cycle map is called the griffiths group and the topological proof here that the griffiths group can be nonzero is the first proof of this fact which does not use hodge theory the proof here gives nonzero torsion elements in the griffiths group whereas griffiths s hodge theoretic proof gives nontorsion elements 13 this topological argument also gives examples of algebraic cycles in the kernel of various related cycle maps where few or no examples were known before thus answering some questions posed by colliot thelene and schoen 8 p 14 37 p 13 colliot thelene asked in particular whether the map ch2 x n h4 x z n is injective for all smooth complex projective varieties x here ch x is the group of codimension i algebraic cycles modulo rational equivalence the first examples where colliot thelene s map is not injective were found by kollar and van geemen see 4 p 135 very recently bloch and esnault found examples defined over number fields 7 over nonalgebraically closed fields k there are other examples of smooth projective varieties xk with ch2 xk n hg4t xk z n not injective due to colliot thelene and sansuc as reinterpreted by salberger see 9 and 8 remark 7 6 1 and parimala and suresh 31 these elements of ch2 xk n are not shown to remain nonzero in ch2 xc n however here our topological method

lichtenbaum tate duality for varieties over p adic fields s lichtenbaum has proved in l1 that there is a nondegenerate pairing pic c x br c br k q z 1 between the picard group and the brauer group of a nonsingular projective curve c over a p adic field k a finite extension of the p adic numbers qp his proof consists of a reduction via explicit cocycle calculations in galois cohomology to a combination of tate duality for group schemes over p adic fields and the autoduality of the jacobian of a smooth curve in this paper we will reconstruct the above duality as a purely formal combination of a generalized form of tate duality over p adic fields and a form of poincar e duality for curves over arbitrary fields of characteristic zero this gives a more conceptual proof of lichtenbaum s result and an analogue in higher dimensions