multicriteria planar location problems abstract given q different objective functions three types of single facility problems are considered lexicographic pareto and max ordering problems after discussing the interrelation between the problem types a complete characterization of lexicographic locations and some instances of pareto and max ordering locations is given the characterizations result in efficient solution algorithms for finding these locations the paper relies heavily on the theory of restricted locations developed by the same authors and can be further extended for instance to multi facility problems with several objectives the proposed approach is more general than previously published results on multicriteria planar location problems and is particularly suited for modelling real world problems

universal deformation rings need not be complete intersections we answer a question of m flach by showing that there is a linear representation of a profinite group whose unrestricted universal deformation ring is not a complete intersection we show that such examples arise in arithmetic in the following way there are infinitely many real quadratic fields f for which there is a mod 2 representation of the galois group of the maximal unramified extension of f whose universal deformation ring is not a complete intersection finally we discuss bounds on the singularities of universal deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups

combinatorial algorithms for some 1 facility median problems in the plane abstract facility location problems in the plane are widely used tools of operations research in modeling real world problems in many of these problems restrictions have to be considered which correspond to regions in which a placement of new locations is forbidden this paper investigates restrictions in 1 facility median problems in the plane we introduce efficient algorithms for problems where the unrestricted problem can be solved in polynomial time the algorithms are of a combinatorial nature and reduce the search for an optimal solution of the restricted problem to a finite number of candidates the assumptions on the model are very weak we only require that the forbidden area is a union of pairwise disjoint convex sets

the classification of endo trivial modules this paper settles the classification of all endo trivial modules for a finite p group by treating the remaining case of a p group having maximal elementary abelian subgroups of rank 2 it is shown that the modules constructed by alperin using relative syzygies provide the complete list of endo trivial modules in this case

universal deformation rings and klein four defect groups in this paper the universal deformation rings of certain modular representations of a finite group are determined the representations under consideration are those which are associated to blocks with klein four defect groups and whose stable endomorphisms are given by scalars it turns out that these universal deformation rings are always subquotient rings of the group ring of a klein four group over the ring of witt vectors

universal deformation rings and generalized quaternion defect groups abstract we determine the universal deformation rings r g v of certain mod 2 representations v of a finite group g which belong to a 2 modular block of g whose defect groups are isomorphic to a generalized quaternion group d we show that for these v a question raised by the author and chinburg concerning the relation of r g v to d has an affirmative answer we also show that r g v is a complete intersection even though r g n v need not be for certain normal subgroups n of g which act trivially on v

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

universal deformation rings and tame blocks let k be an algebraically closed field of positive characteristic and let g be a finite group suppose b is a block of kg of infinite tame representation type we find all finitely generated kg modules v that belong to b and whose endomorphism ring is isomorphic to k and determine the universal deformation ring r g v for each of these modules

derived equivalence classification of algebras of dihedral semidihedral and quaternion type abstract algebras of dihedral semidihedral and quaternion type were introduced by k erdmann as a generalization of blocks of finite groups of tame representation type in a series of papers erdmann gave a description of the morita equivalence classes of these algebras the aim of the present article is to present a classification of algebras of dihedral semidihedral and quaternion type up to derived equivalence as an application we can show that all basic algebras in erdmann s list are actually of tame representation type

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

universal deformation rings and dihedral blocks with two simple modules abstract let k be an algebraically closed field of characteristic 2 and let w be the ring of infinite witt vectors over k suppose g is a finite group and b is a block of kg with a dihedral defect group d such that there are precisely two isomorphism classes of simple b modules we determine the universal deformation ring r g v for every finitely generated kg module v which belongs to b and whose stable endomorphism ring is isomorphic to k the description by erdmann of the quiver and relations of the basic algebra of b is usually only determined up to a certain parameter c which is either 0 or 1 we show that r g v is isomorphic to a subquotient ring of wd for all v as above if and only if c 0 giving an answer to a question raised by the first author and chinburg in this case moreover we prove that c 0 if and only if b is morita equivalent to a principal block

p adic lifting problems and derived equivalences for two derived equivalent k algebras and we introduce a correspondence between o orders reducing to and o orders reducing to we outline how this may be used to transfer properties like uniqueness or non existence of a lift between and as an application we look at tame algebras of dihedral type with two simple modules where most notably we are able to show that among those algebras only the algebras d 0 2a and d 0 2b can actually occur as basic algebras of blocks of group rings of finite groups

logistics distribution centers location problem and algorithm under fuzzy environment distribution centers location problem is concerned with how to select distribution centers from the potential set so that the total relevant cost is minimized this paper mainly investigates this problem under fuzzy environment consequentially chance constrained programming model for the problem is designed and some properties of the model are investigated tabu search algorithm genetic algorithm and fuzzy simulation algorithm are integrated to seek the approximate best solution of the model a numerical example is also given to show the application of the algorithm

using block norms for location modeling in formulating a continuous location model with facilities represented as points in rn e g typically in the plane one must characterize the distance between two points as a function of their coordinates two criteria in selecting a distance function are 1 to obtain good approximations of actual distances and 2 to obtain a mathematical model of the location problem that is easy to solve in this paper we show how a class of norms with polygonal contours called block norms can yield attractive choices as distance functions with respect to these criteria in particular we consider the following relevant properties of block norms they generalize the concepts of rectilinear or city block travel they are dense in the set of all norms they have interesting travel interpretations in the plane they can be expressed as a sum of the absolute values of linear functions they often give better approximations to actual highway distances than the most frequently used family of norms the lp norms and finally they yield linear programming formulations of certain facility location problems i e the weber problem and the rawls problem

α completely positive maps of group systems and krein module representations abstract in this paper we study covariant completely positive maps on group systems we first introduce a notion of completely positive maps of groups into locally c algebras and show that bounded completely positive maps on discrete groups induce completely positive linear maps on group c algebras we establish the covariant ksgns type representation theorem for covariant completely positive maps of group systems into locally c algebras these constructions provide a projective covariant j representation of a group system into a locally c algebra

universal deformation rings and cyclic blocks in this paper we determine the universal deformation rings of certain modular representations of finite groups which belong to cyclic blocks the representations we consider are those for which every endomorphism is stably equivalent to multiplication by a scalar we then apply our results to study the counterparts for universal deformation rings of conjectures about embedding problems in galois theory

error bounds for the approximative solution of restricted planar location problems abstract facility location problems in the plane play an important role in mathematical programming when looking for new locations in modeling real world problems we are often confronted with forbidden regions that are not feasible for the placement of new locations furthermore these forbidden regions may have complicated shapes even if we require them to be convex it may be more useful or even necessary to use approximations of such forbidden regions when trying to solve location problems in this paper we develop error bounds for the approximative solution of restricted planar location problems using the so called sandwich algorithm the number of approximation steps required to achieve a specified error bound is analyzed as examples of these approximation schemes we discuss round norms and polyhedral norms computational tests are also included

geometrical properties of the fermat weber problem abstract this paper is devoted to the fermat weber problem with mixed gauges in order to take into account nonsymmetric distances a geometrical description of the set of solutions is obtained then the concept of metric hull is defined general sufficient conditions are given such that the metric hull of existing facilities intersects the set of optimal locations particularly it is shown that an optimal location can be found in the metric hull in dimension two whatever the gauges are and in dimension n with the rectilinear norm the connection between metric hull convex hull and octagonal hull is studied

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

universal deformation rings and dihedral defect groups let k be an algebraically closed field of characteristic 2 and let w be the ring of infinite witt vectors over k suppose g is a finite group and b is a block of kg with dihedral defect group d which is morita equivalent to the principal 2 modular block of a finite simple group we determine the universal deformation ring r g v for every kg module v which belongs to b and has stable endomorphism ring k it follows that r g v is always isomorphic to a subquotient ring of wd moreover we obtain an infinite series of examples of universal deformation rings which are not complete intersections

localization in multifacility location theory abstract we present two localization theorems for a multifacility location problem involving linear costs and a single norm in a two dimensional space it is shown that optimal locations for all the new facilities can be found in the metric hull of existing facilities whatever the norm is in the polyhedral norm case we study if only finite set of intersections points which belong to the metric hull needs to be considered the result is obtained in particular for the l 1 norm and a counter example is given for a general polyhedral norm