brief equivalence of hybrid dynamical models this paper establishes equivalences among five classes of hybrid systems mixed logical dynamical mld systems linear complementarity lc systems extended linear complementarity elc systems piecewise affine pwa systems and max min plus scaling mmps systems some of the equivalences are established under rather mild additional assumptions these results are of paramount importance for transferring theoretical properties and tools from one class to another with the consequence that for the study of a particular hybrid system that belongs to any of these classes one can choose the most convenient hybrid modeling framework

combinatorial decompositions of a class of rings in this paper we introduce some combinatorial methods for the study of graded rings and use these methods to analyze a class of rings that are of importance in combinatorics our methods are based on a unique representation theorem which is shown section 2 to hold for any graded algebra and which in a sense is an extension of the direct sum decomposition for cohen macaulay algebras this unique representation is reminiscent of a similar result obtained by rees 20 both results yield a decomposition of the algebra however our result differs from that of rees in that it carries more information about the algebra our main results are concerned with a class of rings see section 3 for the definition associated to simplicial complexes there are several natural operations that can be performed on simplicial complexes to obtain new complexes which in turn have an interpretation for the corresponding rings we show sections 4 5 and 6 that our basic decompositions can be transferred during these operations from the original rings to the newly constructed ones the operations studied here are the chain transform or barycentric subdivision section 4 rank selection both for partially ordered sets and for simplicial complexes section s and localization section 6 as we develop these tools we give some applications in section 5 we give a new topological characterization of the cohen macaulay property for partially ordered sets an immediate consequence of which is the rankselection theorem baclawski 2 this characterization is closely related to those obtained by garsia in 14

optimal control for hybrid systems with partitioned state space for hybrid systems where the continuous state space is partitioned by switching manifolds the discrete state changes autonomously if the continuous state hits a manifold recently we introduced a version of the minimum principle for the optimal control of such systems where necessary optimality conditions were provided for the case that a trajectory passes through the intersection of switching manifolds further we introduced an algorithm based on these conditions for computing optimal controls by varying not only the continuous state on switching manifolds based on gradient information but also the sequence of discrete states it was shown that the combinatorial complexity of former computational schemes based on the minimum principle can be avoided since not all possible discrete state sequences need to be analyzed separately in this note theoretical aspects of the algorithm and practical considerations for a successful implementation are discussed in detail further a comparison to existing algorithms for the solution of hybrid optimal control problems with autonomous switching is presented the efficiency of the proposed algorithm is demonstrated by a novel numerical example

alexander duality and stanley depth of multigraded modules abstract we apply miller s theory on multigraded modules over a polynomial ring to the study of the stanley depth of these modules several tools for stanley s conjecture are developed and a few partial answers are given for example we show that taking the alexander duality twice but with different centers is useful for this subject generalizing a result of apel we prove that stanley s conjecture holds for the quotient by a cogeneric monomial ideal

equivalent piecewise affine models of linear hybrid automata in this technical note we examine the relationship between linear hybrid automata lha and piecewise affine pwa systems while a lha is an autonomous non deterministic model a pwa is a deterministic model with inputs through the key idea of modeling the uncertainty associated with lha transitions as input disturbances in a pwa model by extending continuous time pwa models to include the dynamics of discrete states and resets we show in a constructive way that a lha can be equivalently represented as a pwa system where equivalent means that the two systems generate the same trajectories besides filling in a missing theoretical link between the lha modelling framework and the pwa modelling framework the result has the practical advantage of enabling the use of several existing control theoretical tools developed for pwa models to a wider class of hybrid systems

grobner bases of syzygies and stanley depth abstract let f be any free resolution of a finitely generated z n graded module over the polynomial ring k x 1 x n we show that for a suitable term order on f then for 0 p n the initial module of the p 1 th syzygy module z p f p is generated by terms m i e i where the m i are monomials in k x p 1 x n also for a large class of free resolutions f encompassing eliahou kervaire resolutions we show that a grobner basis for z p is given by the boundaries of generators of f p 1 we apply the above to give lower bounds for the stanley depth of the p 1 th syzygy module z p in particular showing it is greater or equal to p 1 we also show that if i is any squarefree ideal in k x 1 x n the stanley depth of i is at least of order 2 n

the price of robustness a robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended under this approach we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes a concern with such an approach is that it might be too conservative in this paper we propose an approach that attempts to make this trade off more attractive that is we investigate ways to decrease what we call the price of robustness in particular we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations an attractive aspect of our method is that the new robust formulation is also a linear optimization problem thus we naturally extend our methods to discrete optimization problems in a tractable way we report numerical results for a portfolio optimization problem a knapsack problem and a problem from the net lib library

hierarchical trajectory optimization for a class of hybrid dynamical systems we discuss trajectory optimization for a class of hybrid systems with a natural hierarchical separation of discrete and continuous dynamics where the continuous dynamics do not change with the discrete state the trajectory optimization problem considered requires that a discrete state sequence and a continuous state trajectory must be both determined to minimize a single cost function such that the discrete state sequence also solves a symbolic planning problem we model this symbolic planning problem as a search on a planning graph and we introduce a family of graphs called lifted planning graphs parametrized by an integer h we define a family of continuous state trajectory optimization problems and associate them with edge costs in the lifted planning graphs next we present an algorithm for finding an optimal solution to the hybrid trajectory optimization problem which includes mapping paths in the lifted planning graphs to discrete state sequences and continuous state trajectories we show that the cost of optimal hybrid trajectories is a nonincreasing function of h and that there exists a finite h for which this cost attains a minimum we illustrate the proposed algorithm with numerical simulation results for two application examples an autonomous mobile vehicle and an autonomous robotic manipulator

a non partitionable cohen macaulay simplicial complex abstract a long standing conjecture of stanley states that every cohen macaulay simplicial complex is partitionable we disprove the conjecture by constructing an explicit counterexample due to a result of herzog jahan and yassemi our construction also disproves the conjecture that the stanley depth of a monomial ideal is always at least its depth

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

stanley depth of complete intersection monomial ideals and upper discrete partitions abstract let i be an m generated complete intersection monomial ideal in s k x 1 x n we show that the stanley depth of i is n m 2 we also study the upper discrete structure for monomial ideals and prove that if i is a squarefree monomial ideal minimally generated by 3 elements then the stanley depth of i is n 1

how to compute the stanley depth of a monomial ideal abstract let j i be monomial ideals we show that the stanley depth of i j can be computed in a finite number of steps we also introduce the fdepth of a monomial ideal which is defined in terms of prime filtrations and show that it can also be computed in a finite number of steps in both cases it is shown that these invariants can be determined by considering partitions of suitable finite posets into intervals

logic based solution methods for optimal control of hybrid systems combinatorial optimization over continuous and integer variables is a useful tool for solving complex optimal control problems of hybrid dynamical systems formulated in discrete time current approaches are based on mixed integer linear or quadratic programming mip which provides the solution after solving a sequence of relaxed linear or quadratic programs mip formulations require the translation of the discrete logic part of the hybrid problem into mixed integer inequalities although this operation can be done automatically most of the original symbolic structure of the problem e g transition functions of finite state machines logic constraints symbolic variables etc is lost during the conversion with a consequent loss of computational performance in this paper we attempt to overcome such a difficulty by combining numerical techniques for solving convex programming problems with symbolic techniques for solving constraint satisfaction problems csp the resulting hybrid solver proposed here takes advantage of csp solvers for dealing with satisfiability of logic constraints very efficiently we propose a suitable model of the hybrid dynamics and a class of optimal control problems that embrace both symbolic and continuous variables functions and that are tailored to the use of the new hybrid solver the superiority in terms of computational performance with respect to commercial mip solvers is shown on a centralized supply chain management problem with uncertain forecast demand

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

optimal control of switched systems based on parameterization of the switching instants this paper presents a new approach for solving optimal control problems for switched systems we focus on problems in which a prespecified sequence of active subsystems is given for such problems we need to seek both the optimal switching instants and the optimal continuous inputs in order to search for the optimal switching instants the derivatives of the optimal cost with respect to the switching instants need to be known the most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state costate stationarity equations the boundary and continuity conditions along with their differentiations this method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed an extension of the method is also applied to problems with internally forced switching examples are shown to illustrate the results in the paper

the number of solutions to linear diophantine equations and multivariate splines etude de la variation du nombre des solutions entieres non negatives de s equations diophantiennes lineaires en fonction des termes non homogenes

dualizing complexes of affine semigroup rings for an affine semigroup ring we construct the dualizing complex in terms of the semigroup and the homology of the face lattice of the polyhedral cone spanned by the semigroup as a consequence there are characterizations of locally cohen macaulay rings buchsbaum rings and cohen macaulay rings as well as serre s condition 9 introduction let s denote a finitely generated submonoid of the additive monoid n n being a positive integer by the affine semigroup ring i s of s over a field x let us denote the subring of x xl xn generated by all monomials xs s e s so one has up to isomorphisms a one to one correspondence between affine semigroup rings and affine varieties given parametrically by monomials now it is of some interest to characterize ring theoretic properties as cohenmacaulay gorenstein etc in terms of the semigroup s a first breakthrough in this direction was done by kempf et al kkms and hochster hol who showed that x s is a cohen macaulay ring provided it is normal moreover the normality of x s is described in terms of s see hol another particular case if s is a simplicial semigroup is treated by goto suzuki and watanabe gsw resp stanley st2 who showed that x s is a cohen macaulay ring if it satisfies the 22 condition of serre note that normality is equivalent to 5y2 and r1 moreover there is a description of the 5 condition in terms of s see theorem 6 3 in general the 52 condition is not sufficient for i s to be a cohen macaulay ring as follows by examples in hol resp th 1 2 on the other side grobner g posed the problem to classify cohen macaulay affine semigroup rings the solution of this problem was done by hoa and trung th related to structural properties of graded rings is a technical tool of the duality theory this is the dualizing complex introduced by grothendieck see ha in recent years it became a helpful tool in commutative algebra in particular its knowledge allows characterizations of cohen macaulay rings locally cohenmacaulay rings buchsbaum rings gorenstein rings and the 5 conditions of received by the editors december 18 1988 1980 mathematics subject classification 1985 revision primary 13h10 secondary 14m05 06a99 1990 american mathematical society 0002 9947 90 1 q0 25 per page

existence theorem for gauge algebra a general gauge action is defined by postulating a minimum of its properties necessary for the existence of loop expansion in the quantum theory the structure of the general gauge algebra is derived from these postulates the proof of existence of the structure functions lacking in previous works is given the question of uniqueness is also completely cleared up some unsolved problems are discussed in connection with the covariant quantization conjecture

stability of depths of symbolic powers of stanley reisner ideals abstract we give a bound for n 0 i from which the depth function depth r i n of the quotient ring by symbolic powers of a squarefree monomial ideal i stabilizes in the unmixed codimension two case we show that depth r i n is a non increasing function of n and use this property to provide a sharp bound for n 0 i

computing the stanley depth in the first chapter some definitions and necessary results from commutative algebra are given also some detail are given which leads us towards the stanley decompositions of multigraded s modules where s is a polynomi ring in n varables over a field f in the end of the chapter we give some important results about the stanledecompositions and stanley conjecture in chapter 2 for the given monomial primary ideals and of s we gave an upper bound for the stanley depth of s which is reached if are irreducible also we showed that stanley conjecture holds for i being some irreducible monomial ideals of s these results are published in our paper 23 for integers 1 n consider the ideal i n s in chapter 3 we gave an upper bound for the stanley depth of the ideal we gave similar upper bounds for the stanley depth of the ideal where in 2 is the square free veronese ideal of degree 2 in n variables these results are from our paper 11

stanley depth of multigraded modules abstract the stanley s conjecture on cohen macaulay multigraded modules is studied especially in dimension 2 in codimension 2 similar results were obtained by herzog soleyman jahan and yassemi as a consequence of our results stanley s conjecture holds in 5 variables

efficient conversion of mixed logical dynamical systems into an equivalent piecewise affine form for hybrid systems described by switched linear difference equations linear threshold conditions automata and propositional logic conditions described in mixed logical dynamical form this note describes two algorithms for transforming such systems into an equivalent piecewise affine form where equivalent means that for the same initial conditions and input sequences the trajectories of the system are identical the proposed techniques exploit ideas from mixed integer programming and multiparametric programming

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

dynamic programming for constrained optimal control of discrete time linear hybrid systems in this paper we study the solution to optimal control problems for constrained discrete time linear hybrid systems based on quadratic or linear performance criteria the aim of the paper is twofold first we give basic theoretical results on the structure of the optimal state feedback solution and of the value function second we describe how the state feedback optimal control law can be constructed by combining multiparametric programming and dynamic programming

depth and stanley depth of symbolic powers of cover ideals of graphs abstract let g be a graph with n vertices and let s k x 1 x n be the polynomial ring in n variables over a field k assume that j g is the cover ideal of g and j g k is its k th symbolic power we prove that the sequences sdepth s j g k k 1 and sdepth j g k k 1 are non increasing and hence convergent suppose that o g denotes the ordered matching number of g we show that for every integer k 2 o g 1 the modules j g k and s j g k satisfy the stanley s inequality we also provide an alternative proof for 9 theorem 3 4 which states that depth s j g k n o g 1 for every integer k 2 o g 1

robust discrete optimization and network flows we propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution and is computationally tractable both practically and theoretically in particular when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation when only the cost coefficients are subject to uncertainty and the problem is a 0 1 discrete optimization problem on n variables then we solve the robust counterpart by solving at most n 1 instances of the original problem thus the robust counterpart of a polynomially solvable 0 1 discrete optimization problem remains polynomially solvable in particular robust matching spanning tree shortest path matroid intersection etc are polynomially solvable we also show that the robust counterpart of an np hard approximable 0 1 discrete optimization problem remains approximable finally we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network

optimal control of a class of hybrid systems we present a modeling framework for hybrid systems intended to capture the interaction of event driven and time driven dynamics this is motivated by the structure of many manufacturing environments where discrete entities termed jobs are processed through a network of workcenters so as to change their physical characteristics associated with each job is a temporal state subject to event driven dynamics and a physical state subject to time driven dynamics based on this framework we formulate and analyze a class of optimal control problems for single stage processes first order optimality conditions are derived and several properties of optimal state trajectories sample paths are identified which significantly simplify the task of obtaining explicit optimal control policies

almost commutative algebra and differential calculus on the quantum hyperplane a notion of almost commutative algebra is given that makes it possible to extend differential geometric ideas associated with commutative algebras in a simple manner to certain classes of noncommutative algebras as an example differential calculus on the n dimensional quantum hyperplane is discussed

stanley depth of squarefree monomial ideals in this paper we answer a question posed by y h shen we prove that if i is an m generated squarefree monomial ideal in the polynomial ring s k x1 xn with k a field then sdepthi n m 2 the proof is inductive and uses the correspondence between a stanley decomposition of a monomial ideal and a partition of a particular poset into intervals established by herzog vladoiu and zheng

local cohomology of modules of covariants abstract let g be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let w u be two finite dimensional representations of g in this paper we compute the local cohomology of u sw g provided a certain relatively weak technical condition is true

nonlinear regulation the piecewise linear approach this paper approaches nonlinear control problems through the use of discrete time piecewise linear systems these are systems whose next state and output maps are both described by pl maps i e by maps which are affine on each of the components of a finite polyhedral partition various results on state and output feedback observers and inverses standard for linear systems are proved for pl systems many of these results are then used in the study of more general both discrete and continuous time systems using suitable approximations

quantum q white noise and a q central limit theorem we establish a connection between the azema martingales and certain quantum stochastic processes with increments satisfyingq commutation relations this leads to a theory ofq white noise onq bialgebras and to a generalization of the fock space representation theorem for white noise on bialgebras in particular quantum azema noise q interpolations between fermion and boson quantum brownian motion and unitary evolutions withq independent multiplicative increments are studied it follows from our results that the azema martingales and theq interpolations are central limits of sums ofq independent identically distributed quantum random variables

on the stanley depth of powers of edge ideals abstract let k be a field and s k x 1 x n be the polynomial ring in n variables over k let g be a graph with n vertices assume that i i g is the edge ideal of g and p is the number of its bipartite connected components we prove that for every positive integer k the inequalities sdepth i k i k 1 p and sdepth s i k p hold as a consequence we conclude that s i k satisfies stanley s inequality for every integer k n 1 also it follows that i k i k 1 satisfies stanley s inequality for every integer k 0 furthermore we prove that if i g is a non bipartite graph or ii at least one of the connected components of g is a tree with at least one edge then i k satisfies stanley s inequality for every integer k n 1 moreover we verify a conjecture of the author in special cases

constructible complexes and recursive division of posets shellability has been extensively studied by many researchers since mcmullen solved the upper bound theorem for convex polytopes there are also some important notions weaker than shellability and in this paper we treat constructibility among these and define a notion of recursively dividable posets which corresponds to the notion of constructible complexes when seeing their face posets also we define a notion of strongly constructible complexes and correspondingly strongly dividable posets by strengthening the conditions and prove that strongly dividable posets are signable this result means that strongly constructible simplicial complexes are partitionable

stanley depths of certain stanley reisner rings abstract let be a simplicial complex on n s k x 1 x n the polynomial ring in n variables over a field k and k s i the stanley reisner ring of with respect to k it is proved that the stanley conjecture holds for k i e sdepth s k depth s k when i has four associated prime ideals it is also obtained that sdepth s k size s i

prime filtrations of monomial ideals and polarizations abstract we show that an arbitrary monomial ideal i is pretty clean if and only if its polarization i p is clean this yields a new characterization of pretty clean monomial ideals in terms of the arithmetic degree and it also implies that a multicomplex is shellable if and only the simplicial complex corresponding to its polarization is non pure shellable we also discuss stanley decompositions in relation to prime filtrations

non commutative resolutions of toric varieties abstract let r be the coordinate ring of an affine toric variety we prove using direct elementary methods that the endomorphism ring end r a where a is the finite direct sum of all isomorphism classes of conic r modules has finite global dimension equal to the dimension of r this gives a precise version and an elementary proof of a theorem of spenko and van den bergh implying that end r a has finite global dimension furthermore we show that end r a is a non commutative crepant resolution if and only if the toric variety is simplicial for toric varieties over a perfect field k of prime characteristic we show that the ring of differential operators d k r has finite global dimension

the ubiquity of smooth hilbert schemes we investigate the geography of hilbert schemes parametrizing closed subschemes of projective space with specified hilbert polynomials we classify hilbert schemes with unique borel fixed points via combinatorial expressions for their hilbert polynomials these expressions naturally lead to an arrangement of nonempty hilbert schemes as the vertices of an infinite full binary tree here we discover regularities in the geometry of hilbert schemes specifically under natural probability distributions on the tree we prove that hilbert schemes are irreducible and nonsingular with probability greater than 0 5

the action of the special orthogonal group on planar vectors integrity bases via a generalization of the symbolic interpretation of molien functions the present article completes the mathematical description initiated in the paper by dhont and zhilinskii 2013 the action of the orthogonal group on planar vectors invariants covariants and syzygies j phys a math theor 46 455202 of the algebraic structures that emerge from the symmetry adapted polynomials in the coordinates of n planar vectors under the action of the so 2 group the set of covariant polynomials contains all the polynomials that transform according to the weight of so 2 and is a free module for but a non free module for the sum of the rational functions of the molien function for covariants describes the decomposition of the ring of invariants or the module of covariants as a direct sum of submodules a method for extracting the generating function for covariants from the comprehensive generating function for all polynomials is introduced the approach allows the direct construction of the integrity basis for the module of covariants decomposed as a direct sum of submodules and gives insight into the expressions for the molien functions found in our earlier paper in particular a generalized symbolic interpretation in terms of the integrity basis of a rational function is discussed where the requirement of associating the different terms in the numerator of one rational function with the same subring of invariants is relaxed

a perspective based convex relaxation for switched affine optimal control abstract we consider the switched affine optimal control problem i e the problem of selecting a sequence of affine dynamics from a finite set in order to minimize a sum of convex functions of the system state we develop a new reduction of this problem to a mixed integer convex program micp based on perspective functions relaxing the integer constraints of this micp results in a convex optimization problem whose optimal value is a lower bound on the original problem value we show that this bound is at least as tight as similar bounds obtained from two other well known micp reductions via conversion to a mixed logical dynamical system and by generalized disjunctive programming and our numerical study indicates it is often substantially tighter using simple integer rounding techniques we can also use our formulation to obtain an upper bound and corresponding sequence of control inputs in our numerical study this bound was typically within a few percent of the optimal value making it attractive as a stand alone heuristic or as a subroutine in a global algorithm such as branch and bound we conclude with some extensions of our formulation to problems with switching costs and piecewise affine dynamics

stanley conjecture in small embedding dimension we show that stanley s conjecture holds for a polynomial ring over a field in four variables in the case of polynomial ring in five variables we prove that the monomial ideals with all associated primes of height two are stanley ideals

simplicity of rings of differential operators in prime characteristic let w be a finite dimensional representation of a linearly reductive group g over a field k motivated by their work on classical rings of invariants levasseur and stafford asked whether the ring of invariants under g of the symmetric algebra of w has a simple ring of differential operators in this paper we show that this is true in prime characteristic indeed if r is a graded subring of a polynomial ring over a perfect field of characteristic p 0 and if the inclusion r hookrightarrow s splits then d k r is a simple ring in the last section of the paper we discuss how one might try to deduce the characteristic zero case from this result as yet however this is a subtle problem and the answer to the question of levasseur and stafford remains open in characteristic zero http www luc ac be research algebra 1991 mathematics subject classification 16s32 16g60 13a35

local cohomology with support in ideals of maximal minors and sub maximal pfaffians abstract we compute the gl equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix as well as of those with support in the ideal of 2 n 2 n pfaffians of a 2 n 1 2 n 1 generic skew symmetric matrix as an application we characterize the cohen macaulay modules of covariants for the action of the special linear group sl g on g m the main tool we develop is a method for computing certain ext modules based on the geometric technique for computing syzygies and on matlis duality