on the depth of the tangent cone and the growth of the hilbert function for a d dimensional cohen macaulay local ring r mn we study the depth of the associated graded ring of r with respect to an rm primary ideal i in terms of the vallabrega valla conditions and the length of it jit where j is a j minimal reduction of i and t 1 as a corollary we generalize sally s conjecture on the depth of the associated graded ring with respect to a maximal ideal to rm primary ideals we also study the growth of the hilbert function

avoidability of circular formulas abstract clark has defined the notion of n avoidance basis which contains the avoidable formulas with at most n variables that are closest to be unavoidable in some sense the family c i of circular formulas is such that c 1 a a c 2 a b a b a b c 3 a b c a b c a b c a b c and so on for every i n the n avoidance basis contains c i clark showed that the avoidability index of every circular formula and of every formula in the 3 avoidance basis and thus of every avoidable formula containing at most 3 variables is at most 4 we determine exactly the avoidability index of these formulas

degree and birationality of multi graded rational maps we give formulas and effective sharp bounds for the degree of multi graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties we also extend the jacobian dual criterion to the multi graded setting our approach is based on the study of blow up algebras including syzygies of the ideal generated by the defining polynomials of the rational map a key ingredient is a new algebra that we call thesaturated special fiber ring which turns out to be a fundamental tool to analyze the degree of a rational map we also provide a very effective birationality criterion and a complete description of the equations of the associated rees algebra of a particular class of plane rational maps

grothendieck serre formula and bigraded cohen macaulay rees algebras abstract the grothendieck serre formula for the difference between the hilbert function and hilbert polynomial of a graded algebra is generalized for bigraded standard algebras this is used to get a similar formula for the difference between the bhattacharya function and bhattacharya polynomial of two m primary ideals i and j in a local ring a m in terms of local cohomology modules of rees algebras of i and j the cohomology of a variation of the kirby mehran complex for bigraded rees algebras is studied which is used to characterize the cohen macaulay property of bigraded rees algebra of i and j for two dimensional cohen macaulay local rings

when are multidegrees positive abstract let k be an arbitrary field p p k m 1 k k p k m p be a multiprojective space over k and x p be a closed subscheme of p we provide necessary and sufficient conditions for the positivity of the multidegrees of x as a consequence of our methods we show that when x is irreducible the support of multidegrees forms a discrete algebraic polymatroid in algebraic terms we characterize the positivity of the mixed multiplicities of a standard multigraded algebra over an artinian local ring and we apply this to the positivity of mixed multiplicities of ideals furthermore we use our results to recover several results in the literature in the context of combinatorial algebraic geometry

fitting ideals and multiple points of surface parameterizations given a birational parameterization f of an algebraic surface s the purpose of this paper is toinvestigate the sets of points on s whose preimage consists in kor more points counting multiplicities these points are described explicitly interms of fitting ideals of some graded parts of the symmetric algebraassociated to the parameterization f to obtain this description weshow that the degree and dimension of a fiber could be computed bycomparing the drop of rank of two explicit representation matrices associated to f

on dejean s conjecture over large alphabets the maximal exponent of a non empty finite word is the ratio of its length to its period dejean 1972 conjectured that for any n 5 there exists an infinite word over n letters with no factor of its exponent larger than n n 1 we prove that this conjecture is true for n 33

subword complexity and power avoidance abstract we begin a systematic study of the relations between subword complexity of infinite words and their power avoidance among other things we show that the thue morse word has the minimum possible subword complexity over all overlap free binary words and all 7 3 power free binary words but not over all 7 3 power free binary words the twisted thue morse word has the maximum possible subword complexity over all overlap free binary words but no word has the maximum subword complexity over all 7 3 power free binary words if some word attains the minimum possible subword complexity over all square free ternary words then one such word is the ternary thue word the recently constructed 1 2 bonacci word has the minimum possible subword complexity over all symmetric square free ternary words

sensitivity based methods for convergence acceleration of iterative algorithms abstract a new method to accelerate the convergence of iterative schemes for the numerical integration of systems of partial differential equations has been developed the basic idea is that the residual at a grid point depends on the values of the solution vector at the neighboring grid points used in the local discretization approximation thus the new acceleration method is based on the sensitivity of the future residual to the change in the solution vector at the neighboring grid points with the objective to minimize the future residual the result is a set of optimum iterative relaxation parameters for the entire flow field or for each individual grid line the method is easy to implement in the existing codes we have applied it to a finite difference code for two dimensional incompressible navier stokes equations test cases involve laminar and turbulent flows with severe grid clustering and flow separation the results are compared with those of a basic explicit runge kutta rk time stepping iterative algorithm and with the implicit residual smoothing 1rs and the distributed minimal residual dmr acceleration techniques the new acceleration scheme is shown to be superior to these methods especially on highly clustered grids

a geometric theory of the buchsbaum rim multiplicity abstract we develop a self contained theory of a generalized buchsbaum rim multiplicity based on elementary abstract algebraic geometry we define the multiplicity as the sum of certain natural intersection numbers and recover the traditional definition in terms of the leading coefficient of the polynomial that gives the appropriate lengths we interpret the partial sums as the multiplicities in certain closely associated cases we prove an additivity formula in the rings and relate constancy to a generalized notion of reductions we establish a polar multiplicity formula and a mixed multiplicity formula and we determine when each of the four types of multiplicities vanish we end by generalizing the height inequality for maximal minors and a celebrated theorem of boger

finiteness of cousin cohomologies he notion of the cousin complex of a module was given by sharp in 1969 it wasn t known whether its cohomologies are finitely generated until recently in 2001 dibaei and tousi showed that the cousin cohomologies of a finitely generated a module m are finitely generated if the base ring a is local has a dualizing complex m satisfies serre s s 2 condition and is equidimensional in the present article the author improves their result he shows that the cousin cohomologies of m are finitely generated if a is universally catenary all the formal fibers of all the localizations of a are cohen macaulay the cohen macaulay locus of each finitely generated a algebra is open and all the localizations of m are equidimensional as a consequence of this he gives a necessary and sufficient condition for a noetherian ring to have an arithmetic macaulayfication

bass numbers in the graded case a invariant formulas and an analogue of faltings annihilator theorem abstract let r n 0rn be a positively graded commutative noetherian ring a graded r module is called indecomposable respectively injective if it is indecomposable respectively injective in the category of graded r modules a graded prime ideal of r is called irrelevant if it contains r n 0rn let m n z mn be a finitely generated graded r module the first main result of the paper presents a refinement of the theory of the bass numbers of m with respect to an irrelevant graded prime ideal p of r this refinement is related to the siting in terms of degrees of indecomposable injective direct summands having associated prime p of terms in the minimal injective resolution of m the second main result has a consequence which can be described in terms of the castelnuovo regularity r of m any indecomposable injective direct summand having irrelevant associated prime of any term in the minimal injective resolution of m must vanish in all degrees greater than r results of this type provide uniform upper bounds for the degrees of influence of such irrelevant primes and these bounds are exploited in the later sections of the paper to a generalize known a invariant formulas due to e hyry and n v trung b to throw new light on the fact that for an ideal c 0 of r0 the sequence grademn c 0 n is ultimately constant and c to describe the ultimate constant value of that sequence by means of an analogue of faltings annihilator theorem for local cohomology

acceleration of iterative algorithms for euler equations of gasdynamics of the grid size the solution is considered converged when the maximum absolute residual is one order of magnitude smaller than the truncation error the convergence history for the stream function vorticity formulation is similar the residual of the linear poisson equation for the stream function is always of machine accuracy and the residual of the vorticity transport equation reduces quadratically convergence is obtained after six iterations for the biharmonic equation convergence takes six iterations and is also quadratic the solutions in terms of the wall skin friction distribution are presented in fig 2 the results for the three formulations are in excellent agreement and correlate well with the skin friction distribution presented by briley 8 next the model problem of a separated flow in a symmetrical diffuser introduced by inoue 9 is examined the diffuser problem is solved using the stream function vorticity and the biharmonic formulation the inlet and outlet boundary of the diffuser are at x 1 0 and x 3 0 respectively the shape factor of the diffuser wall a 0 089 the centerline is located at y 1 0 and the reynolds number based on this reference length and the free stream velocity is r 6250 the inflow conditions provide the initial conditions for the entire flowfield convergence is quadratic and machine zero is reached in 6 7 iterations the solutions in terms of the wall skin friction distribution are shown in fig 3 the results for the two formulations are in excellent agreement and correlate well with inoue s results except for the outflow conditions in fig 3 the solution for the higher reynolds number r 12500 with a larger separation region is also presented the biharmonic equation is the most efficient of the three formulations in terms of cpu time and storage requirements the biharmonic program is more than two times faster and requires more than a factor two less memory than the streamfunction vorticity program the primitive variable method is the slowest among the three formulations and it puts severe demands on the computer storage requirements in this case the bandwidth is 0 67v and this coupled with the increase in the number of variables results in more than a four fold increase in storage and cpu time as compared to the biharmonic program conclusions three formulations of the two dimensional navier stokes equations are solved numerically using newton s method and a direct solution routine for banded matrices the fully implicit solution techniques use second order central differencing for all the terms and are shown to be reliable and to provide quadratic convergence the biharmonic formulation is most efficient in terms of cpu time and memory without loss of accuracy finally while it is well known that iterative methods line overrelaxation or adi for biharmonic equations have very slow rates of convergence the present study using direct solvers indicates that the biharmonic formulation is the most recommended

acceleration of iterative algorithms on highly clustered grids a family of new methods has been developed to accelerate the convergence rate of iterative algorithms for obtaining a steady state solution as an asymptotic limit of an unsteady second order partial differential equation or a system of such equations it was assumed that a central differencing has been used for spatial discretization the new acceleration methods are based on the sensitivity of the future residual at every grid point to the change in the solution vector components at the neighboring grid points used in the local discretization approximation the acceleration parameters introduced in the methods have been optimized with the objective to minimize the future global residual the new sensitivity based methods have been applied to finite difference codes for twoand three dimensional laminar incompressible flow navier stokes equations two dimensional turbulent incompressible flow navier stokes equations and two dimensional compressible flow euler equations the new sensitivity based acceleration methods demonstrated superior performance in all test cases that involved severe grid clustering and grid nonorthogonality and included laminar and turbulent flows with closed and open flow separation

multigraded regularity reduction vectors and postulation vectors abstract we relate the set of complete reduction vectors of a z s graded admissible filtration of ideals f with the set of multigraded regularities of g f we prove reg g f reg r f we establish a relation between the sets of complete reduction vectors of f and postulation vectors of f under some cohomological conditions

the largest non vanishing degree of graded local cohomology modules abstract this paper gives characterizations for the largest non vanishing degree of the local cohomology modules of a graded ring s in terms of a reduction of s and of the associated graded ring of an ideal i in terms of any reduction of i as a consequence this invariant can be computed explicitly for the associated graded ring of any ideal generated by a d sequence and of ideals having small analytic deviation

castelnuovo mumford regularity with respect to multigraded ideals abstract in this article we extend a previous definition of castelnuovo mumford regularity for modules over an algebra graded by a finitely generated abelian group our notion of regularity is based on maclagan and smith s definition and is extended first by working over any commutative base ring and second by considering local cohomology with support in an arbitrary finitely generated graded ideal b obtaining for each b a b regularity region the first extension provides a natural approach for working with families of sheaves or of graded modules while the second opens new applications even in the more restrictive framework where castelnuovo mumford was defined before us there were only very partial results on estimates for the shifts in a minimal graded free resolution from the castelnuovo mumford regularity we prove sharp estimates in our general framework and this is one of our main advances we provide tools to deduce information on the graded betti numbers from the knowledge of regions where the local cohomology with support in a given graded ideal vanishes conversely vanishing of local cohomology with support in any graded ideal is deduced from the shifts in a free resolution and the local cohomology of the polynomial ring the flexibility of treating local cohomology with respect to any b opens up new possibilities for passing information we provide new persistence results for the vanishing of local cohomology that extend the fact that weakly regular implies regular in the classical case and we give sharp estimates for the regularity of a truncation of a module in the last part we present a result on hilbert functions for multigraded polynomial rings which provides a simple proof of the generalized grothendieck serre formula

asymptotic behaviour of arithmetically cohen macaulay blow ups this paper addresses problems related to the existence of arithmetic macaulayfications of projective schemes let y be the blow up of a projective scheme x projr along the ideal sheaf of i r it is known that there are embeddings y projk i e c for c d i e 1 where d i denotes the maximal generating degree of i and that there exists a cohen macaulay ring of the form k i e c if and only if h 0 y oy k h i y oy 0 for i 1 dimy 1 y is equidimensional and cohen macaulay cutkosky and herzog asked when there is a linear bound on c and e ensuring that k i e c is a cohen macaulay ring we obtain a surprising compelte answer to this question namely that under the above conditions there are well determined invariants and e0 such that k i e c is cohen macaulay for all c d i e and e e0 our approach is based on recent results on the asymptotic linearity of the castelnuovo mumford regularity of ideal powers we also investigate the existence of a cohen macaulay rees algebra of the form r i e ct which provides an arithmetic macaulayfication for x if r has negative a invariant we prove that such a cohen macaulay rees algebra exists if and only if oy ox r i oy 0 for i 0 y is equidimensional and cohen macaulay especially these conditions imply the cohen macaulayness of r i e ct for all c d i e and e e0 the above results can be applied to obtain several new classes of cohen macaulay algebras

studies of numerical methods for the plane navier stokes equations abstract in the present report and in our most recent work we have concentrated on three basic formulations primitive variables vorticity stream function and stream function biharmonic formulation with each such formulation there are a variety of difference equations that could be used and then there are a large number of iteration schemes that could be employed to solve these difference equations the ultimate goal of course is to find the best combination of all these techniques

hilbert polynomials of multigraded filtrations of ideals abstract hilbert functions and hilbert polynomials of z s graded admissible filtrations of ideals f n n z s such that r f n is finite for all n z s are studied conditions are provided for the hilbert function h f n r f n and the corresponding hilbert polynomial p f n to be equal for all n n s a formula for the difference h f n p f n in terms of local cohomology of the extended rees algebra of f is proved which is used to obtain sufficient linear relations analogous to the ones given by huneke and ooishi among coefficients of p f n so that h f n p f n for all n n s a theorem of rees about joint reductions of the filtration i r j s r s z is generalised for admissible filtrations of ideals in two dimensional cohen macaulay local rings necessary and sufficient conditions are provided for the multi rees algebra of an admissible z 2 graded filtration f to be cohen macaulay

dejean s conjecture holds for n 30 abstract we extend carpi s results by showing that dejean s conjecture holds for n 30

on arithmetic macaulayfication of noetherian rings the rees algebra is the homogeneous coordinate ring of a blowing up the present paper gives a necessary and sufficient condition for a noetherian local ring to have a cohen macaulay rees algebra a noetherian local ring has a cohen macaulay rees algebra if and only if it is unmixed and all the formal fibers of it are cohen macaulay as a consequence of it we characterize a homomorphic image of a cohen macaulay local ring for non local rings this paper gives only a sufficient condition by using it however we obtain the affirmative answer to sharp s conjecture that is a noetherian ring having a dualizing complex is a homomorphic image of a finite dimensional gorenstein ring

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

bounding hilbert coefficients of parameter ideals abstract let r m be a noetherian local ring of dimension d 0 and depth r d 1 let q be a parameter ideal of r in this paper we derive uniform lower and upper bounds for the hilbert coefficient e i q under certain assumptions on the depth of associated graded ring g q for 2 i d we show that 1 e i q 0 provided depth g q d 2 and 2 e i q r h m d 1 r provided depth g q d 1 it is proved that e 3 q 0 further we obtain a necessary condition for the vanishing of the last coefficient e d q as a consequence we characterize the vanishing of e 2 q our results generalize 5 theorem 3 2 and 11 corollary 4 5

hilbert polynomials of non standard bigraded algebras this paper investigates hilbert polynomials of bigraded algebras which are generated by elements of bidegrees 1 0 d 1 1 ldots d r 1 where d 1 ldots d r are non negative integers the obtained results can be applied to study rees algebras of homogeneous ideals and their diagonal subalgebras

supporting degrees of multi graded local cohomology modules for a finitely generated graded module m over a positively graded commutative noetherian ring r the second author established in 1999 some restrictions which can be formulated in terms of the castelnuovo regularity of m or the so called a invariant of m on the supporting degrees of a graded indecomposable graded injective direct summand with associated prime ideal containing the irrelevant ideal of r of any term in the minimal graded injective resolution of m earlier in 1995 t marley had established connections between finitely graded local cohomology modules of m and local behaviour of m across proj r the purpose of this paper is to present some multi graded analogues of the above mentioned work

asymptotic vanishing of homogeneous components of multigraded modules and its applications abstract in this article we give a condition on the vanishing of finitely many homogeneous components which must imply the asymptotic vanishing for multigraded modules we apply our result to multi rees algebras of ideals as a consequence we obtain a result on normality of monomial ideals which extends and improves the results of reid roberts and vitulli 14 singla 16 and sarkar and verma 15

gorenstein graded rings associated to ideals let a be a noetherian local ring with the maximal ideal m and dim a d let i 6 a be an ideal in a and s hta i this article studies the question of when the associated graded ring g i i 0 i i is gorenstein to state our result we set up some notation let be an integer such that d and let j be a reduction of i generated by elements a1 a2 a we denote by rj i the reduction number of i with respect to j the analytic spread of i is i dim a m a g i then s i we always assume that the generating set a1 a2 a of j is a basic generating set for j in the sense of aberbach huneke and trung aht which means that jiaq is a reduction of iaq for all q v i with i hta q here let v i be a set of prime ideals in a containing i and ji a1 a2 ai for 0 i by aht 7 2 there always exists a basic generating set for j if the field a m is infinite let

proof of dejean s conjecture for alphabets with 5 6 7 8 9 10 and 11 letters abstract axel thue proved that overlapping factors could be avoided in arbitrarily long words on a two letter alphabet while on the same alphabet square factors always occur in words longer than 3 francoise dejean stated an analogous result for three letter alphabets every long enough word has a factor which is a fractional power with an exponent at least 7 4 and there exist arbitrary long words in which no factor is a fractional power with an exponent strictly greater than 7 4 the number 7 4 is called the repetition threshold of the three letter alphabets thereafter she proposed the following conjecture the repetition threshold of the k letter alphabets is equal to k k 1 except in the particular cases k 3 where this threshold is 7 4 and k 4 where it is 7 5 for k 4 this conjecture was proved by j j pansiot 1984 in this paper we give a computer aided proof of dejean s conjecture for several other values 5 6 7 8 9 10 and 11

overlap free words and finite automata abstract a method to represent certain words on a binary alphabet by shorter words on a larger alphabet is introduced we prove that overlap free words are represented by the words of a rational language several consequences are derived concerning the density function of the set of overlap free words on a binary alphabet and the prolongability of overlap free words in particular efficient algorithms are obtained computing the density function of the set of overlap free words on a binary alphabet testing whether a word on a binary alphabet is overlap free and computing its depth

when does depth stabilize early on in this paper we study graded ideals i in a polynomial ring s such that the numerical function k depth s ik is constant we show that if i the rees algebra of i is cohen macaulay ii the cohomological dimension of i is not larger than the projective dimension of s i and iii the k algebra generated by some homogeneous generators of i is a direct summand of s then depth s ik is constant all the ideals with constant depth function discovered by herzog and vladoiu in 11 satisfy the criterion given above in the contest of square free monomial ideals there is a chance that a converse of the previous fact holds true

fast iterative algorithms based on optimized explicit time stepping abstract the generalized nonlinear minimal residual gnlmr method is shown to consistently accelerate and stabilize iterative algorithms for solving nonlinear problems by using the optimized explicit multistepping the examples presented in this paper illustrate the beneficial effects of the optimized multistep algorithm on the computational efficiency and the convergence rate as applied to several nonlinear problems in fluid dynamics the significant reduction in computing time when using the multiple optimized acceleration factors is only negligibly weighed down by the computation costs due to the requirements for additional computer storage

on the number of dejean words over alphabets of 5 6 7 8 9 and 10 letters we give lower bounds on the growth rate of dejean words i e minimally repetitive words over a k letter alphabet for 5 k 10 put together with the known upper bounds we estimate these growth rates with the precision of 0 005 as a consequence we establish the exponential growth of the number of dejean words over a k letter alphabet for 5 k 10

multigraded regularity a invariant and the minimal free resolution abstract in recent years two different multigraded variants of castelnuovo mumford regularity have been developed namely multigraded regularity defined by the vanishing of multigraded pieces of local cohomology modules and the resolution regularity vector defined by the multi degrees in a minimal free resolution in this paper we study the relationship between multigraded regularity and the resolution regularity vector our method is to investigate multigraded variants of the usual a invariant this in particular provides an effective approach to examining the vanishing of multigraded pieces of local cohomology modules with respect to different multigraded ideals

reduction numbers and multiplicities of multigraded structures the aim of this paper is twofold first we want to relate the cohen macaulay property of r i i to the cohen macaulay propa 1 r erty of the ordinary rees rings r i j s 1 r then we determine a j the multiplicities of multi rees and form rings in particular we ask when minimal multiplicity occurs for i s s i s i the multi rees algebras r i s r i i 1 r a r a behave to some extent like the ordinary rees algebras r i in particua lar the cohen macaulay property of such multi rees algebras can be characterized in terms of conditions on the local cohomology of the form ring gr i including the condition a gr i 0 where a gr i denotes a a a w x the a invariant of gr i see hhr 1 the a invariant has in certain a w cases been related to the reduction exponent r i of the ideal i see t2 x gh moreover the cohen macaulay property of r i implies that of a w x r i for all r 1 the converse is not true in general see hhr1 but a r in certain applications the assumption r i is cohen macaulay can be a

the structure of sally modules towards a theory of non cohen macaulay cases abstract sally modules of m primary ideals in a generalized cohen macaulay and or buchsbaum local ring a m are investigated the structure theorems of sally modules of rank one given by goto et al 2008 gno1 gno2 in the case where a is a cohen macaulay local ring are generalized

cohomological properties of non standard multigraded modules in this paper we study some cohomological properties of non standard multigraded modules and veronese transforms of them among others numerical characters we study the generalized depth of a module and we see that it is invariant by taking a veronese transform we prove some vanishing theorems for the local cohomology modules of a multigraded module as a corollary of these results we get that the depth of a veronese module is asymptotically constant

curvilinear base points local complete intersection and koszul syzygies in biprojective spaces let i be a bigraded ideal in the bigraded polynomial ring k s u t v assume that i has codimension 2 then z v i c p 1 p 1 is a finite set of points we prove that if z is a local complete intersection then any syzygy of the f i vanishing at z and in a certain degree range is in the module of koszul syzygies this is an analog of a recent result of cox and schenck 2003

multigraded rings diagonal subalgebras and rational singularities we study f rationality and f regularity in diagonal subalgebras of multigraded rings and use this to construct large families of rings that are f rational but not f regular we also use diagonal subalgebras to construct rings with divisor class groups that are finitely generated but not discrete in the sense of danilov