group preference aggregation methods employed in ahp an evaluation and an intrinsic process for deriving members weightages abstract the analytic hierarchy process ahp is one of the popular and powerful techniques for decision making a detailed survey of the literature has revealed that there exists no formal evaluation of the group preference aggregation methods currently employed in ahp this paper provides such an evaluation using well established social choice axioms which govern the process of combining individual opinions to obtain a single group opinion the geometric mean method gmm and the weighted arithmetic mean method wamm are the two methods evaluated it is shown using counter examples that the gmm does not always satisfy the pareto optimality axiom which is one of the prominent and widely accepted social choice axioms this finding is significant as the gmm has been the most commonly used method in ahp for combining individual opinions to form a group opinion the other method viz wamm has satisfied all the axioms except the independence of irrelevant alternatives axiom in order to use the wamm one has to find the weightages importance to be assigned to the members of the group this is often a difficult task especially so if the group is large as in the case of public policy decisions and when judgements are elicited through the use of questionnaires these situations need an objective method to derive members weightages but only a few studies are available in the literature to address such a situation we propose a simple and intuitively appealing eigenvector based method to intrinsically determine the weightages for group members using their own subjective opinions the superiority of the proposed method over the previous methods is brought out in the paper

linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

ideals with stable betti numbers componentwise linear ideals were introduced earlier to generalize the result that the stanley reisner ideal i of a simplicial complex has a linear resolution if and only if its alexander dual is cohen macaulay it turns out that i is componentwise linear if and only if is sequentially cohen macaulay in this paper we discuss betti number properties of componentwise linear ideals let i be a graded ideal of a polynomial ring s and gin i the generic initial ideal of i with respect to the reverse lexicographic term order on s our main result is that i and gin i admit the same graded betti numbers if and only if i is componentwise linear for the proof of this fact we describe some properties of the betti diagram of a generic initial ideal combinatorial implications for shifted complexes will also be discussed

multigraded castelnuovo mumford regularity we develop a multigraded variant of castelnuovo mumford regularity motivated by toric geometry we work with modules over a polynomial ring graded by a finitely generated abelian group as in the standard graded case our definition of multigraded regularity involves the vanishing of graded components of local cohomology we establish the key properties of regularity its connection with the minimal generators of a module and its behavior in exact sequences for an ideal sheaf on a simplicial toric variety x we prove that its multigraded regularity bounds the equations that cut out the associated subvariety we also provide a criterion for testing if an ample line bundle on x gives a projectively normal embedding

generalized castelnuovo mumford regularity for affine kac moody algebras abstract the graded modules over noncommutative algebras often have minimal free resolutions of infinite length resulting in infinite castelnuovo mumford regularity in kang et al 2010 6 we introduced a generalized notion of castelnuovo mumford regularity to overcome this difficulty in this paper we compute the generalized castelnuovo mumford regularity for integrable highest weight representations of all affine kac moody algebras it is shown that the generalized regularity depends only on the type and rank of algebras and the level of representations

a diagonal bound for cohomological postulation numbers of projective schemes abstract let x be a projective scheme over a field k and let f be a coherent sheaf of o x modules we show that the cohomological postulation numbers f i of f e g the ultimate places at which the cohomological hilbert functions n dim k h i x f n h f i n start to be polynomial for n 0 are polynomially bounded in terms of the cohomology diagonal h f i i i 0 dim f of f as a consequence we obtain that there are only finitely many different cohomological hilbert functions h f i if f runs through all coherent sheaves of o x modules with fixed cohomology diagonal in order to prove these results we extend the regularity bound of bayer and mumford computational algebraic geometry and commutative algebra proc cortona 1991 cambridge univ press 1993 pp 1 48 from graded ideals to graded modules moreover we prove that the castelnuovo mumford regularity of the dual f of a coherent sheaf of o p k r modules f is polynomially bounded in terms of the cohomology diagonal of f

the information encoded in initial ideals we consider homogeneous ideals i and the initial ideal in i for the revlex order first we give a sequence of invariants computed from i giving better and better approximations to the initial ideal and ending in an equivalent description then we apply this to different settings in algebraic geometry to understand what information is encoded in the generic initial ideal of the ideal of a projective scheme we also consider the higher initial ideals as defined in a paper by fl0ystad in particular we show that giving the generic higher initial ideal of a space curve is equivalent to giving the generic initial ideal of a linked curve

recent improvements in the complexity of the effective nullstellensatz we bring up to date the estimates on the complexity of the effective nullstellensatz and the membership problem

a combinatorial proof of the eisenbud goto conjecture for monomial curves and some simplicial semigroup rings abstract we will give a pure combinatorial proof of the eisenbud goto conjecture for arbitrary monomial curves moreover we will show that the conjecture holds for certain simplicial affine semigroup rings

empirical properties of group preference aggregation methods employed in ahp theory and evidence abstract we study various methods of aggregating individual judgments and individual priorities in group decision making with the ahp the focus is on the empirical properties of the various methods mainly on the extent to which the various aggregation methods represent an accurate approximation of the priority vector of interest we identify five main classes of aggregation procedures which provide identical or very similar empirical expressions for the vectors of interest we also propose a method to decompose in the ahp response matrix distortions due to random errors and perturbations caused by cognitive biases predicted by the mathematical psychology literature we test the decomposition with experimental data and find that perturbations in group decision making caused by cognitive distortions are more important than those caused by random errors we propose methods to correct the systematic distortions

saturation and castelnuovo mumford regularity abstract let i be a homogeneous ideal of the polynomial ring k x 0 x n where k is an arbitrary field avoiding the construction of a minimal graded free resolution of i we provide effective methods for computing the castelnuovo mumford regularity of i that also compute other cohomological invariants of k x 0 x n i

cohomology of projective schemes from annihilators to vanishing we provide bounds on the castelnuovo mumford regularity in terms of defining equations by using elements that annihilates some cohomology modules inspired by works of miyazaki nagel schenzel and vogel the elements in these annihilators are provided either by liaison or by tight closure theories our results hold in any characteristic

the regularity of a toric variety abstract we give a method for computing the degrees of the minimal syzygies of a toric variety by means of combinatorial techniques indeed we complete the explicit description of the minimal free resolution of the associated semigroup algebra using the simplicial representation of koszul homology which appeared in a campillo and c marijuan 1991 sem theor nombres bordeaux3 249 260 as an application we obtain an algorithm for computing the castelnuovo mumford regularity of a projective toric variety this regularity is explicitly bounded by means of the semigroup generators which parametrize the variety

circuit complexity proof complexity and polynomial identity testing the ideal proof system we introduce a new and natural algebraic proof system whose complexity measure is essentially the algebraic circuit size of nullstellensatz certificates this enables us to exhibit close connections between effective nullstellensatze proof complexity and algebraic circuit complexity in particular we show that any super polynomial lower bound on any boolean tautology in our proof system implies that the permanent does not have polynomial size algebraic circuits vnp vp we also show that super polynomial lower bounds on the number of lines in polynomial calculus proofs imply the permanent versus determinant conjecture note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound our proof system helps clarify the relationships between previous algebraic proof systems in doing so we highlight the importance of polynomial identity testing pit in proof complexity in particular we use pit to illuminate ac0 p frege lower bounds which have been open for nearly 30 years with no satisfactory explanation as to their apparent difficulty finally we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity using the algebraic structure of our proof system we propose a novel route to such lower bounds although such lower bounds remain elusive this proposal should be contrasted with the difficulty of extending ac0 p circuit lower bounds to ac0 p frege lower bounds

remarks on the analytic hierarchy process the analytic hierarchy process ahp is flawed as a procedure for ranking alternatives in that the rankings produced by this procedure are arbitrary this paper provides a brief review of several areas of operational difficulty with the ahp and then focuses on the arbitrary rankings that occur when the principle of hierarchic composition is assumed this principle requires that the weights on the higher levels of a hierarchy can be determined independently of the weights on the lower levels virtually all of the published examples of the use of the ahp to evaluate alternatives relative to a set of criteria have assumed this principle the key to correcting this flaw is the synthesis of the ahp with the concepts of multiattribute utility theory

almost regular sequences and betti numbers let be a simplicial complex and s the shifted simplicial complex of as defined by kalai it is shown that the stanley reisner ideals i and i s have the same extremal betti numbers this theorem is the squarefree analogue of the corresponding result by bayer charalambous and popescu which asserts that a graded ideal and its generic ideal have the same extremal betti numbers our theorem implies well known results by kalai according to which and s have the same simplicial homology and is cohen macaulay if and only if its shifted complex is cohen macaulay

bounds for the castelnuovo mumford regularity of modules we establish bounds for the castelnuovo mumford regularity of a finitely generated graded module and its symmetric powers in terms of the degrees of the generators of the module and the degrees of their relations we extend to modules and improve the known bounds for homogeneous ideals in a polynomial ring established by galligo giusti caviglia and sbarra

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

castelnuovo mumford regularity postulation numbers and reduction numbers abstract suppose g is a standard graded ring over an infinite field we obtain a sharp lower bound for the regularity of g in terms of the postulation number the depth and the dimension of g we present a class of examples in dimension 1 where the postulation number is 0 and the regularity of g can take on any value between 1 and the embedding codimension of g suppose g gr m r is the associated graded ring of a cohen macaulay local ring r m we compute the regularity the reduction number and the postulation number of g and consider the relationship among these invariants in the case where dim g grade g 1 a precise description is known as to how these integers are related we consider the case where dim g grade g 2 and prove that if dim g grade g 2 then reg g max p g dim g 1 r m where p g is the postulation number of g and r m is the reduction number of m

curves on normal surfaces we study locally cohen macaulay space curves lying on normal surfaces we prove some theorems on the behaviour of the cohomology functions and initial ideals of such space curves which give a basic distinction between those curves and curves lying on non normal surfaces a natural step in the study and classification of space curves is to classify the hilbert functions of space curves or more generally cohomology functions of space curves a closely related approach is to study the somewhat more refined invariants the generic initial ideal and higher initial ideal and classify which of them can occur for smooth integral or locally cohen macaulay space curves given a space curve c on a surface s p one may do descending biliaison on the surface s and end up with a curve x minimal in its biliaison class on the surface s on the other hand one might consider a surface t of minimal degree such that c may be linked s t to a curve y in these cases it will often happen thatx or y will have non reduced components the invariants of c may be deduced from the invariants of x or y hence it is natural to try to get an understanding of the invariants of curves on s with nonreduced components here it turns out that there is a basic distinction between when s is normal and when s is not normal the reason is that if one has a reduced curve d on s then a multiplicity n structure on d in s is uniquely determined if s is normal while if s is not normal there may exist many in fact infinitely many multiplicity n structures on d in s for c a curve on a surface s of degree s there is a basic sequence 0 l os oc 0 1 where l is the ideal sheaf of c in s this also gives rise to a sequence 0 os s 4 l s 4 c 0 2 in both cases l and l s 4 are torsion free rank one sheaves on s with local projective dimension one as op3 modules we prove the following theorem 2 1 let s p be a normal surface of degree s and l a torsion free rank one sheaf on s with local projective dimension one as op3 module let the minimal generators of h l n zh l n occur in degrees n0 n1 nr then nj nj 1 s 2 for j 1 r and the regularity of l is nr s 2 received by the editors july 4 1996 and in revised form september 22 1997 2000 mathematics subject classification primary 14h50

vanishing theorems a theorem of severi and the equations defining projective varieties in 1903 severi s proved that if x x c p are smooth surfaces whose union is the complete intersection of r 2 hypersurfaces of degrees di dr 2 then hypersurfaces of degree k z di r cut out a complete linear series on x cf sr xiii 9 8 the purpose of this paper is first of all to show that elementary arguments using the kodaira vanishing theorem lead to a simple variant of severi s statement theorem 7 below which extends it in serval directions more importantly we hope to convince the reader that this result has a surprising number of applications to questions involving the equations defining projective varieties consider to begin with a smooth complex projective variety x c pr of dimension n and codimension e r n in this setting our theorem asserts the vanishing of the higher cohomology of suitable twists of the ath power of the ideal sheaf yx of x in apr proposition 1 assume that x is cut out scheme theoretically in p by hypersurfaces of degrees d1 d2 d then h p r k 0 for i 1 provided k ad d2 de r note that only the degrees of the first e codim x pr defining equations come into play here when n 2 and a 1 this is a consequence of severi s

on equations defining arithmetically cohen macaulay schemes i in this paper we continue the study of the simplest arithmetically cohen macauly cm for short schemes cf 111 and 12 i we describe the hilbert function of an arbitrary cm scheme in p of given degree e lying on a hypersurface of given degree k and such that its general curvilinear section has minimal genus proposition 1 3 we present a geometric construction of all reduced irreducible codimension two cm varieties whose general curvilinear sections have minimal genus with respect to degree theorem 1 7 we also prove the following assertion essentially due to severi there exist smooth nonspecial cm curves of degree n and genus g in p iff r y1 g r d t y 1 2 proposition 1 6 any such curve has minimal genus with respect to degree in the last section we consider d dimensional cm schemes of degree e mvd j jed in p whose general curvilinear sections have minimal genus these are the most accessible cm schemes cf 12 i we prove that the corresponding part of the hilbert scheme is a connected smooth variety naturally birationally equivalent to nd symemvd p provided v d 2 theorem 2 6 this was known for m v d 2 3 1 cf 9 our proof relies on an explicit description of the infinitesimal deformations of a thick subspace of arbitrary codimension see 2 4 we work over an algebraically closed field of characteristic 0 by the genus of a cm curve we mean its arithmetic genus

someinverseproblemsforellipticmatriceswith zero diagonal abstract we present explicit formulae which allow us to construct elliptic matrices with zero diagonal from a given spectrum as well as from the complete interlacing system

conditions for generic initial ideals to be almost reverse lexicographic abstract let i be a homogeneous artinian ideal in a polynomial ring r k x 1 x n over a field k of characteristic 0 we study an equivalent condition for the generic initial ideal gin i with respect to reverse lexicographic order to be almost reverse lexicographic as a result we show that the moreno socias conjecture implies the froberg conjecture and the pardue conjecture and for the case codim i 3 we show that r i has the strong lefschetz property if and only if gin i is almost reverse lexicographic finally we give a positive partial answer to the moreno socias conjecture and hence to the pardue conjecture

a stratification of the hilbert scheme of points in the projective plane this result is analogous to a theorem of iarrobino and brian on on the stratification of the punctual hilbert scheme where a different hilbert function defined by initial forms instead is used however cp i l thm 2 9 p 26 the proof essentially makes use of the smoothness of h a cp f p 517 and a general result of bialynicki birula on the action of the multiplicative group on a smooth projective scheme as already was done in a study of the homology of h a by ellingsrud and stromme this method is applied here to a study of h a around points which are invariant under the borel group in the appendix a method for computing the dimension of h is represented which rests on iarrobino s description of the tangent space to certain points of h a and the theorem of bialynicki birula finally i would like to mention that hirschowitz has outlined a general program for studying stratifications including those defined by hilbert functions cp h p 175 but the result obtained here can at best be regarded only as the most simple perhaps non trivial example

an experiment on the consistency of aggregated comparison matrices in ahp the analytic hierarchy process can be used for group decision making by aggregating individual judgments or individual priorities the most commonly used aggregation methods are the geometric mean method and the weighted arithmetic mean method while it is known that the weighted geometric mean comparison matrix is of acceptable consistency if all individual comparison matrices are of acceptable consistency this paper addresses the following question under what conditions would an aggregated geometric mean comparison matrix be of acceptable consistency if some or all of the individual comparison matrices are not of acceptable consistency using monte carlo simulation results indicate that given a sufficiently large group size consistency of the aggregate comparison matrix is guaranteed regardless of the consistency measures of the individual comparison matrices if the geometric mean is used to aggregate this result implies that consistency at the aggregate level is a non issue in group decision making when group size exceeds a threshold value and the geometric mean is used to aggregate individual judgments this paper determines threshold values for various dimensions of the aggregated comparison matrix

lefschetz properties and hyperplane arrangements abstract in this article we study the weak and strong lefschetz properties and the related notion of almost revlex ideal in the non artinian case proving that several results known in the artinian case hold also in this more general setting we then apply the obtained results to the study of the jacobian algebra of hyperplane arrangements

castelnuovo mumford regularity of seminormal simplicial affine semigroup rings abstract we show that the eisenbud goto conjecture holds for homogeneous seminormal simplicial affine semigroup rings moreover we prove an upper bound for the castelnuovo mumford regularity in terms of the dimension which is similar as in the normal case finally we compute explicitly the regularity of full veronese rings

direct methods for primary decomposition leti be an ideal in a polynomial ring over a perfect field we given new methods for computing the equidimensional parts and radical ofi for localizingi with respect to another ideal and thus for finding the primary decomposition ofi our methods rest on modern ideas from commutative algebra and are direct in the sense that they avoid the generic projections used by hermann 1926 and all others until now

koszul homology and extremal properties of gin and lex for every homogeneous ideal i in a polynomial ring r and for every p dim r we consider the koszul homology h i p r i with respect to a sequence of p of generic linear forms the koszul betti number ijp r i is by definition the dimension of the degree j part of h i p r i in characteristic 0 we show that the koszul betti numbers of any ideal i are bounded above by those of the gin revlex gin i of i and also by those of the lex segment lex i of i we show that ijp r i ijp r gin i iff i is componentwise linear and that and ijp r i ijp r lex i iff i is gotzmann we also investigate the set gins i of all the gin of i and show that the koszul betti numbers of any ideal in gins i are bounded below by those of the gin revlex of i on the other hand we present examples showing that in general there is no j is gins i such that the koszul betti numbers of any ideal in gins i are bounded above by those of j

finiteness of hilbert functions and bounds for castelnuovo mumford regularity of initial ideals bounds for the castelnuovo mumford regularity and hilbert coefficients are given in terms of the arithmetic degree if the ring is reduced or in terms of the defining degrees from this it follows that there exists only a finite number of hilbert functions associated with reduced algebras over an algebraically closed field with a given arithmetic degree and dimension a good bound is also given for the castelnuovo mumford regularity of initial ideals which depends neither on term orders nor on the coordinates and holds for any field

group preference aggregation in the multiplicative ahp the model of the group decision process and pareto optimality abstract a recent paper has focused awareness on group aggregation procedures in the ahp showing that geometric mean aggregation violates the desirable social choice axiom of pareto optimality we show that this violation can be attributed to the representation used to model the group decision process thereby questioning the legitimacy of the pareto optimality axiom we furthermore propose a geometric mean group aggregation procedure which satisfies all the social choice axioms suggested

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

asymptotic resurgences for ideals of positive dimensional subschemes of projective space abstract recent work of ein lazarsfeld smith and hochster huneke raised the containment problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal bocci harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings with a focus on zero dimensional subschemes of projective space here we take the first steps toward extending this work to higher dimensional subschemes we introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals i of smooth subschemes generalizing what is done in bocci and harbourne 2010 5 we apply these bounds to ideals of unions of general lines in p n we also pose a nagata type conjecture for symbolic powers of ideals of lines in p 3

a q analogue of the distance matrix of a tree we consider a q analogue of the distance matrix called the q distance matrix of an unweighted tree and give formulae for the inverse and the determinant which generalize the existing formulae for the distance matrix we obtain the smith normal form of the q distance matrix of a tree the relationship of the q distance matrix with the laplacian matrix leads to q analogue of the laplacian matrix of a tree some of whose properties are also given we study another matrix related to the distance matrix the exponential distance matrix and show its relationship with the q laplacian and the q distance matrix a formula for the determinant of the q distance matrix of a weighted tree is also given

linear normality of general linear sections and some graded betti numbers of 3 regular projective schemes in this paper we study graded betti numbers of any nondegenerate 3 regular algebraic set x in a projective space pn more concretely via generic initial ideals gins method we mainly consider tailing betti numbers whose homological index is at least codim x pn for this purpose we first introduce a key definition nd 1 property which provides a suitable ground where one can generalize the concepts such as being nondegenerate or of minimal degree from the case of varieties to the case of more general closed subschemes and give a clear interpretation on the tailing betti numbers next we recall basic notions and facts on gins theory and we analyze the generation structure of the reverse lexicographic rlex gins of 3 regular nd 1 subschemes as a result we present exact formulae for these tailing betti numbers which connect them with linear normality of general linear sections of x with a linear subspace of dimension at least codim x pn finally we consider some applications and related examples

test multiplier and invariant ideals abstract this paper gives an explicit formula for the multiplier ideals and consequently for the log canonical thresholds of any gl v gl w invariant ideal in s sym v w where v and w are vector spaces over a field of characteristic 0 this characterization is done in terms of a polytope constructed from the set of young diagrams corresponding to the schur modules generating the ideal our approach consists in computing the test ideals of some invariant ideals of s in positive characteristic namely we compute the test ideals and so the f pure thresholds of any sum of products of determinantal ideals not all the invariant ideals are as the latter not even in characteristic 0 but they are up to integral closure and this is enough to reach our goals the results concerning the test ideals are obtained as a consequence of general results holding true in a special situation within such framework fall determinantal objects of a generic matrix as well as of a symmetric matrix and of a skew symmetric one similar results are thus deduced for the gl v invariant ideals in sym sym 2 v and in sym 2 v monomial ideals also fall in this framework thus we recover howald s formula for their multiplier ideals and more generally hara yoshida s formula for their test ideals in the proof we introduce the notion of floating test ideals a property that in a sense is satisfied by ideals defining schemes with the nicest possible singularities as will be shown products of determinantal ideals and by passing to characteristic 0 ideals generated by a single schur module have this property

borel fixed ideals and reduction number abstract the aim of this paper is to study the relationship between the reduction number and borel fixed ideals in all characteristics especially it is shown that r r i r r i lex where i lex denotes the unique lex segment ideal whose hilbert function is equal to that of i this solves a recent question by conca

aggregation of analytic hierarchy process models based on similarities in decision makers preferences abstract the analytic hierarchy process is widely used in both individual and group decision making environments in this paper we investigate its applicability to model a specific class of decentralized decision problems where many decision makers take individual subjective decisions using locally available information in such subjective decision making environments it is neither possible nor appropriate to use group preference aggregation techniques to model the problem as a single group decision problem an approach to identify homogeneous subgroups of decision makers based on similarities in preferences and to aggregate preferences within each subgroup is proposed this approach is validated using employment preferences of 70 subjects modeled using the analytic hierarchy process