small and strong formulations for unions of convex sets from the cayley embedding there is often a significant trade off between formulation strength and size in mixed integer programming when modeling convex disjunctive constraints e g unions of convex sets adding auxiliary continuous variables can sometimes help resolve this trade off however standard formulations that use such auxiliary continuous variables can have a worse than expected computational effectiveness which is often attributed precisely to these auxiliary continuous variables for this reason there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables we introduce a technique to construct formulations without these detrimental continuous auxiliary variables to develop this technique we introduce a natural non polyhedral generalization of the cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding we then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints in particular we show it can recover and generalize all known strong formulations without continuous auxiliary variables

base change and a problem of serre we establish a version of level lowering for mod p galois representations arising from the reductions of representations associated to hilbert modular forms in particular we show that level lowering can be easily achieved if one replaces the base field with a suitable solvable extension this is often enough for applications to proving the modularity of p adic representations

convex extensions and envelopes of lower semi continuous functions we define a convex extension of a lower semi continuous function to be a convex function that is identical to the given function over a pre specified subset of its domain convex extensions are not necessarily constructible or unique we identify conditions under which a convex extension can be constructed when multiple convex extensions exist we characterize the tightest convex extension in a well defined sense using the notion of a generating set we establish conditions under which the tightest convex extension is the convex envelope then we employ convex extensions to develop a constructive technique for deriving convex envelopes of nonlinear functions finally using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of x y over a rectangle and prove that the extensions theory provides convex relaxations that are much tighter than the relaxation provided by the classical outer linearization of bilinear terms

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

convex envelopes generated from finitely many compact convex sets we consider the problem of constructing the convex envelope of a lower semi continuous function defined over a compact convex set we formulate the envelope representation problem as a convex optimization problem for functions whose generating sets consist of finitely many compact convex sets in particular we consider nonnegative functions that are products of convex and component wise concave functions and derive closed form expressions for the convex envelopes of a wide class of such functions several examples demonstrate that these envelopes reduce significantly the relaxation gaps of widely used factorable relaxation techniques

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

bounds for the size of radar arrays etablissement de bornes ameliorees superieure et inferieure pour la taille des tableaux radars formes de matrices rectangulaires a elements 0 et 1

two term disjunctions on the second order cone balas introduced disjunctive cuts in the 1970s for mixed integer linear programs several recent papers have attempted to extend this work to mixed integer conic programs in this paper we study the structure of the convex hull of a two term disjunction applied to the second order cone and develop a methodology to derive closed form expressions for convex inequalities describing the resulting convex hull our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex possibly nonlinear valid inequalities that correspond to these linear inequalities we identify and study the cases where these valid inequalities can equivalently be expressed in conic quadratic form and where a single inequality from this family is sufficient to describe the convex hull in particular our results on two term disjunctions on the second order cone generalize related results on split cuts by modaresi kilinc and vielma and by andersen and jensen

the taylor wiles construction and multiplicity one wiles proof 17 of the modularity of semistable elliptic curves over q relies on a construction of taylor and wiles 16 showing that certain hecke algebras are complete intersections these hecke algebras are defined by considering the action of hecke operators on spaces of modular forms of minimal level or equivalently on homology groups or jacobians of modular curves taylor and wiles proceed roughly speaking by patching algebras arising from forms of different levels one of the deep results used in their construction was the fact that the homology of the modular curve becomes a free module of rank two over the hecke algebra upon localization at certain maximal ideals this result known as a multiplicity one result is a generalization of a theorem of mazur 11 its proof relied on the q expansion principle of deligne rapoport and katz and the comparison of mod betti and de rham cohomologies see sect 2 1 of 17 multiplicity one was thought to be a crucial ingredient of the taylor wiles construction as well as other parts of wiles proof the purpose of this paper is to explain how to alter the arguments of 16 and 17 so that multiplicity one results are a byproduct rather than an ingredient the key conceptual change underlying this improvement is the following rather than prove that after localization the hecke algebra can be identified with the universal deformation ring of a mod galois representation we prove that the homology of the modular curve is a free module over this deformation ring to carry this out we change the taylor wiles construction1 by 1 patching the modules as well as the algebras and 2 applying the auslander buchsbaum

two dimensional synchronization patterns for minimum ambiguity a number of closely related combinatorial problems corresponding to specific assumptions about the type of time frequency sequence which may be appropriate in a particular application are formulated in terms of square or rectangular arrays of dots with appropriate constraints on the two dimensional correlation function the current state of knowledge concerning each of these problems is summarized it is hoped that more general constructions may be found leading to larger families of solutions as well as better computational algorithms for finding individual solutions which may lie outside of the general families

on serre s conjecture for mod l galois representations over totally real fields in 1987 serre conjectured that any mod two dimensional irre ducible odd representation of the absolute galois group of the rationals came from a modular form in a precise way we present a generalisation of this conjecture to 2 dimensional representations of the absolute galois group of a totally real eld where is unramied the hard work is in formulating an analogue of the weight part of serre s conjecture serre furthermore asked whether his conjecture could be rephrased in terms of a mod langlands philosophy using ideas of emerton and vign eras we formulate a mod local global principle for the group d where d is a quaternion algebra over a totally real eld split above and at 0 or 1 innite places and show how it implies the conjecture

finding meaning in error terms four decades ago mikio sato and john tate predicted the shape of probability distributions to which certain error terms in number theory conform their prediction known as the sato tate conjecture has been verified for an important class of cases thanks to the recent work of laurent clozel michael harris and richard taylor 3 and of michael harris nicholas shepherd barron and richard taylor 16 combined with richard taylor s most recent 50 which establishes this advance in our understanding

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

a convex analysis perspective on disjunctive cuts we treat with tools from convex analysis the general problem of cutting planes separating a point from a closed convex set p crucial for this is the computation of extreme points in the so called reverse polar set introduced by e balas in 1979 in the polyhedral case this enables the computation of cuts that define facets of p we exhibit three equivalent optimization problems to compute such extreme points one of them corresponds to selecting a specific normalization to generate cuts we apply the above development to the case where p is the closed convex hull of a union and more particularly a union of polyhedra case of disjunctive cuts we conclude with some considerations on the design of efficient cut generators the paper also contains an appendix reviewing some fundamental concepts of convex analysis

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

explicit convex and concave envelopes through polyhedral subdivisions in this paper we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper rectangle these envelopes are obtained by identifying polyhedral subdivisions of the hyper rectangle over which the envelopes can be constructed easily in particular we use these techniques to derive in closed form the concave envelopes of concave extendable supermodular functions and the convex envelopes of disjunctive convex functions

multiplicites modulaires et representations de rm gl sb 2 mathbf z sb p et de rm gal overline mathbf q sb p mathbf q sb p en ell p appendice par guy henniart sur l unicite des types pour rm gl sb 2

deriving convex hulls through lifting and projection we consider convex hull descriptions for certain sets described by inequality constraints over a hypercube and propose a lifting and projection technique to construct them the general procedure obtains the convex hulls as an intersection of semi infinite families of linear inequalities each derived using lifting techniques that are interpreted using convexification tools we demonstrate that differentiability and concavity of certain perturbation functions help reduce the number of inequalities needed for this characterization each family of inequalities yields a few linear nonlinear constraints fully characterized in the space of the original problem variables when the projection problems are amenable to a closed form solution in particular we illustrate the complete procedure by constructing the convex hulls of the subsets of a compact hypercube defined by the constraints x 1 b 1 x 2 b 2 ge x 3 and x 1 x 2 b 2 le x 3 where b 1 b 2 ge 1 as a consequence we obtain a closed form description of the convex hull of the bilinear equality x 1x 2 x 3 in the presence of variable bounds as an intersection of a few linear and nonlinear inequalities this explicit characterization hitherto unknown can improve relaxation techniques for factorable functions which utilize this equality to relax products of functions with known relaxations

on icosahedral artin representations ii we prove that some new infinite families of odd two dimensional icosahedral representations of the absolute galois group of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i are modular and hence satsify the artin conjecture we also give an account of work of ramakrishna on lifting mod l galois representations to characteristic zero

general constructions of optimal variable weight optical orthogonal codes variable weight optical orthogonal code ooc was introduced by yang for multimedia optical cdma systems with multiple quality of service qos requirements in this paper four general constructions for optimal cyclic packings and optimal variable weight oocs are presented many new infinite classes of optimal v w 1 q oocs are obtained new infinite classes of optimal v w 1 q oocs for w 4 are easily obtained by the constructions

modularity of 2 adic barsotti tate representations we prove a modularity lifting theorem for two dimensional 2 adic potentially barsotti tate representations this proves hypothesis h of khare wintenberger and completes the proof of serre s conjecture the main new ingredient is a classification of connected finite flat group schemes over rings of integers of finite extensions of 2

new bounds for the sizes of radar arrays improved upper bounds for the size of radar arrays are found by using the window method new general constructions and computer searched examples are presented the radar array problem is generalized to include arrays with sidelobes greater than 1

on serre s conjecture for 2 dimensional mod p representations of gal q q we prove the existence in many cases of minimally ramified p adic lifts of 2 dimensional continuous odd absolutely irreducible mod p representations of the absolute galois group of q it is predicted by serre s conjecture that such representations arise from newforms of optimal level and weight

strong valid inequalities for orthogonal disjunctions and bilinear covering sets in this paper we derive a closed form characterization of the convex hull of a generic nonlinear set when this convex hull is completely determined by orthogonal restrictions of the original set although the tools used in this construction include disjunctive programming and convex extensions our characterization does not introduce additional variables we develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed we demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed integer polyhedral sets and finding the convex hull of certain mixed pure integer bilinear sets we then extend the utility of the convexification tool to relaxing nonconvex inequalities which are not naturally disjunctive by providing sufficient conditions for establishing the convex extension property over the non negative orthant we illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non negative orthant although we illustrate our results primarily on bilinear covering sets they also apply to more general polynomial covering sets for which they yield new tight relaxations

a polyhedral branch and cut approach to global optimization a variety of nonlinear including semidefinite relaxations have been developed in recent years for nonconvex optimization problems their potential can be realized only if they can be solved with sufficient speed and reliability unfortunately state of the art nonlinear programming codes are significantly slower and numerically unstable compared to linear programming software in this paper we facilitate the reliable use of nonlinear convex relaxations in global optimization via a polyhedral branch and cut approach our algorithm exploits convexity either identified automatically or supplied through a suitable modeling language construct in order to generate polyhedral cutting planes and relaxations for multivariate nonconvex problems we prove that if the convexity of a univariate or multivariate function is apparent by decomposing it into convex subexpressions our relaxation constructor automatically exploits this convexity in a manner that is much superior to developing polyhedral outer approximators for the original function the convexity of functional expressions that are composed to form nonconvex expressions is also automatically exploited root node relaxations are computed for 87 problems from globallib and minlplib and detailed computational results are presented for globally solving 26 of these problems with baron 7 2 which implements the proposed techniques the use of cutting planes for these problems reduces root node relaxation gaps by up to 100 and expedites the solution process often by several orders of magnitude

a precise correspondence between lift and project cuts simple disjunctive cuts and mixed integer gomory cuts for 0 1 programming we establish a precise correspondence between lift and project cuts for mixed 0 1 programs simple disjunctive cuts intersection cuts and mixed integer gomory cuts the correspondence maps members of one family onto members of the others it also maps bases of the higher dimensional cut generating linear program cglp into bases of the linear programming relaxation it provides new bounds on the number of facets of the elementary closure and on the rank of the standard linear programming relaxation of the mixed 0 1 polyhedron with respect to the above families of cutting planes

on unions and dominants of polytopes a well known result on unions of polyhedra in the same space gives an extended formulation in a higher dimensional space whose projection is the convex hull of the union here in contrast we study the unions of polytopes in different spaces giving a complete description of the convex hull without additional variables when the underlying polytopes are monotone the facets are described explicitly generalizing results of hong and hooker on cardinality rules and an efficient separation algorithm is proposed these results are based on an explicit representation of the dominant of a polytope and an algorithm for the separation problem for the dominant for non monotone polytopes both the dominant and the union are characterized we also give results on the unions of polymatroids both on disjoint ground sets and on the same ground set generalizing results of conforti and laurent

a systematic construction for radar arrays the radar array problem arises from the need to design frequency hopping sequences with small out of phase autocorrelations it assumes the reflected signals have negligible doppler shifts so the correlations are calculated along the time axis only in this correspondence a systematic construction for radar arrays is provided by means of homogeneous uniform difference matrices a systematic construction for properly centered permutation matrices a special kind of homogeneous uniform difference matrices is also provided which partially solves the open problems posed by zhang and tu

how to convexify the intersection of a second order cone and a nonconvex quadratic a recent series of papers has examined the extension of disjunctive programming techniques to mixed integer second order cone programming for example it has been shown by several authors using different techniques that the convex hull of the intersection of an ellipsoid mathcal e e and a split disjunction l x j x j u le 0 l xj xj u 0 with l u l

strong formulations of robust mixed 0 1 programming we introduce strong formulations for robust mixed 0 1 programming with uncertain objective coefficients we focus on a polytopic uncertainty set described by a budget constraint for allowed uncertainty in the objective coefficients we show that for a robust 0 1 problem there is an tight linear programming formulation with size polynomial in the size of an tight linear programming formulation for the nominal 0 1 problem we give extensions to robust mixed 0 1 programming and present computational experiments with the proposed formulations

convex programming for disjunctive convex optimization given a finite number of closed convex sets whose algebraic representation is known we study the problem of finding the minimum of a convex function on the closure of the convex hull of the union of those sets we derive an algebraic characterization of the feasible region in a higher dimensional space and propose a solution procedure akin to the interior point approach for convex programming

geometric orthogonal codes of size larger than optical orthogonal codes the class of geometric orthogonal codes gocs was introduced by doty and winslow 2016 for more robust macrobonding in dna origami they observed that gocs are closely related to optical orthogonal codes oocs it is possible for gocs to have size greater than oocs of corresponding parameters due to slightly more relaxed constraints on correlations however the existence of gocs exceeding the size of optimal oocs of corresponding parameters has never been demonstrated this paper gives the first infinite family of gocs of size greater than optimal oocs

modularity of certain potentially barsotti tate galois representations where is the reduction of p these results were subject to hypotheses on the local behavior of p at x i e the restriction of p to a decomposition group at x and to irreducibility hypotheses on 1p in this paper we build on the methods of 461 451 and 121 and relax the hypotheses on local behavior in particular we treat certain adic representations which are not semistable at x but potentially semistable we do this using results of 61 generalizing a theorem of ramakrishna 321 see fontaine mazur 22 131 for a slightly different point of view the results in 61 show that certain potentially barsotti tate deformation problems are smooth allowing us to define certain universal deformations for with the necessary galoistheoretic properties to apply wiles method to carry out the proof that these deformations are indeed realized in the cohomology of modular curves i e that the universal deformation rings are hecke algebras we need to identify the corresponding cohomology groups and prove they have the modular theoretic properties needed to apply wiles method as in 151 and 121 the identification is made by matching local behavior of automorphic representations and galois representations via the local langlands correspondence together with fontaine s theory at the prime we work directly with cohomology of modular curves instead of

chva tal closures for mixed integer programming problems chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality this viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems the basic ingredient in these cutting plane proofs is that for a polyhedronp and integral vectorw if max wx x p wx integer t thenwx t is valid for all integral vectors inp we consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that bar wx be integer for some other integral vector bar w the cutting plane proofs thus obtained may be seen either as an abstraction of gomory s mixed integer cutting plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by balas and jeroslow our main result is that for a given polyhedronp the set of vectors that satisfy every cutting plane forp with respect to a specified subset of integer variables is again a polyhedron this allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron analogous to the process of repeatedly taking chvatal closures in the integer programming case these results are illustrated with a number of examples from combinatorial optimization our work can be seen as a continuation of that of nemhauser and wolsey on mixed integer cutting planes

a branch and cut method for 0 1 mixed convex programming we generalize the disjunctive approach of balas ceria and cornuejols 2 and devevlop a branch and cut method for solving 0 1 convex programming problems we show that cuts can be generated by solving a single convex program we show how to construct regions similar to those of sherali and adams 20 and lovasz and schrijver 12 for the convex case finally we give some preliminary computational results for our method

construction of some families of 2 dimensional crystalline representations we construct explicitly some analytic families of etale modules which give rise to analytic families of 2 dimensional crystalline representations as an application of our constructions we verify some conjectures of breuil on the reduction modulo p of those representations and extend some results of deligne edixhoven fontaine and serre on the representations arising from modular forms

different transformations for solving non convex trim loss problems by minlp in the present paper trim loss problems often named the cutting stock problem connected to the paper industry are considered the problem is to cut out a set of product paper rolls from raw paper rolls such that the cost function including the trim loss as well as the costs for the over production is minimized the problem is non convex due to certain bilinear constraints the problem can however be transformed into linear or convex form the resulting transformed problems can thereafter be solved as mixed integer linear programming problems or convex mixed integer non linear programming problems the linear and convex formulations are attractive from a formal point of view since global optimal solutions to the originally non convex problem can be obtained however as the examples considered will show the numerical efficiency of the solutions from the different transformed formulations varies considerably an example based on a trim optimization problem encountered daily at a finnish paper converting mill is finally presented in order to demonstrate differences in the numerical solutions

solving mixed integer bilinear problems using milp formulations in this paper we examine a mixed integer linear programming reformulation for mixed integer bilinear problems where each bilinearterm involves the product of a nonnegative integer variable and a nonnegative continuous variable this reformulation is obtained by first replacing a general integer variable with its binary expansion and then using mccormick envelopes to linearize the resulting product of continuous and binary variables we present the convex hull of the underlying mixed integer linear set the effectiveness of this reformulation and associated facet defining inequalities are computationally evaluated on five classes of instances

simultaneous convexification of bilinear functions over polytopes with application to network interdiction we study the simultaneous convexification of graphs of bilinear functions g k x y y intercal a k x over x in ensuremath xi x in 0 1 n e x geq f and y in delta m y in r m 1 intercal y leq 1 we propose a constructive procedure to obtain a linear description of the convex hull of the resulting set this procedure can be applied to derive convex and concave envelopes of certain bilinear functions to study unary expansions of integer variables in mixed integer bilinear sets and to obtain convex hulls of sets with complementarity constraints exploiting the structure of xi the procedure naturally yields stronger linearizations for bilinear terms in a variety of practical settings in particular we demonstrate the effectiveness of the approach by strengthening the traditional dual formulation of network interdiction problems and report encouraging preliminary numerical results

lifted inequalities for 0 mathord 1 mixed integer bilinear covering sets in this paper we study 0 mathord 1 mixed integer bilinear covering sets we derive several families of facet defining inequalities via sequence independent lifting techniques we then show that these sets have a polyhedral structure that is similar to that of a certain fixed charge single node flow set as a result we also obtain new facet defining inequalities for the single node flow set that generalize well known lifted flow cover inequalities from the integer programming literature

a local to global principle for deformations of galois representations given an absolutely irreducible galois representation ge gln k e a number eld k a nite eld of characteristic l 2 and a nite set of places q of e containing all places above l and1 and all where ramies there have been dened many functors representing strict equivalence classes of deformations of such a representation e g by mazur or wiles in 15 or 26 with various conditions on the behaviour of the deformations at the places in q and with the condition that the deformations are unramied outside q those functors are known to be representable for as above our goal is to present a rather general class of global deformation functors that satisfy local deformation conditions and to investigate for those under what conditions the global deformation functor is determined by the local deformation functors we will give precise conditions under which the local functors for all places in q are sucient to describe the global functor rst in a coarse form then in a rened form using auxiliary primes as done by taylor and wiles in 24 this has several consequences the strongest is that one can derive ring theoretic results for the universal deformation space by mazur if one uses results of diamond and wiles c f 11 and 26 and if one has a good understanding of all local situations furthermore it is easier to understand what happens under increasing the ramication

a new framework to relax composite functions in nonlinear programs in this paper we devise new relaxations for composite functions which improve the prevalent factorable relaxations without introducing additional variables by exploiting the inner function structure we outer approximate inner functions using arbitrary under and over estimators and then convexify the outer function over a polytope p which models the ordering relationships between the inner functions and their estimators and utilizes bound information on the inner functions as well as on the estimators we show that there is a subset q of p with significantly simpler combinatorial structure such that the separation problem of the graph of the outer function over p is polynomially equivalent via a fast combinatorial algorithm to that of its graph over q we specialize our study to consider the product of two inner functions with one non trivial underestimator for each inner function for the corresponding polytope p we show that there are eight valid inequalities besides the four mccormick inequalities which improve the factorable relaxation finally we show that our results generalize to simultaneous convexification of a vector of outer functions

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

hecke stability and weight 1 modular forms the galois representations associated to weight 1 eigenforms over bar mathbb f p are remarkable in that they are unramified at p but the effective computation of these modular forms presents challenges one complication in this setting is that a weight 1 cusp form over bar mathbb f p need not arise from reducing a weight 1 cusp form over bar mathbb q in this article we propose a unified hecke stability method for computing spaces of weight 1 modular forms of a given level in all characteristics simultaneously our main theorems outline conditions under which a finite dimensional hecke module of ratios of modular forms must consist of genuine modular forms we conclude with some applications of the hecke stability method that are motivated by the refined inverse galois problem

on a conjecture of conrad diamond and taylor we prove a conjecture of conrad diamond and taylor on the size of certain deformation rings parametrizing potentially barsotti tate ga lois representations to achieve this we extend results of breuil and m ezard classifying galois lattices in semistable representations in terms of strongly divisible modules to the potentially crystalline case in hodge tate weights 0 1 we then use these strongly divisible modules to compute the desired deformation rings as a corollary we obtain new results on the modularity of potentially barsotti tate representations

golomb rectangles there are a number of communications problems involving two dimensional arrays with constrained correlation functions golomb s two classes of radar arrays and sonar arrays are examined improved bounds optimal cases and computational procedures are reported

technical note on generating cutting planes from combinatorial disjunctions this paper develops a technique for generating deep valid inequalities based on combinatorial disjunctions that state that at least p out of some q p constraints must be satisfied this technique provides a general framework which subsumes existing methods for generating cutting planes based on such disjunctions it has potential applications for example in multiple choice and generalized lattice point programming problems including mixed integer zero one programming problems and the extreme point mathematical programming problem