subdivisions and local h vectors in part i a general theory of f vectors of simplicial subdivisions or triangulations of simplicial complexes is developed based on the concept of lo cal h vector as an application we prove that the h vector of a cohen macaulay complex increases under quasi geometric subdivision thus establishing a spe cial case of a conjecture of kalai and this author techniques include com mutative algebra homological algebra and the intersection homology of toric varieties in part ii we extend the work of part i to more general situations first a formal generalization of subdivision is given based on incidence alge bras special cases are then developed in particular one based on subdivisions of eulerian posets and involving generalized h vectors other cases deal with kazhdan lusztig polynomials ehrhart polynomials and a q analogue of eule rian posets many applications and examples are given throughout department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address rstan math mit edu this content downloaded from 157 55 39 210 on sat 30 jul 2016 05 59 16 utc all use subject to http about jstor org terms

on the ideal theory of graphs we study algebras defined by finite sets g m1 mq of monomials of a polynomial ring r there are two basic algebras i k g k m1 mq the k subalgebra of r spanned by the mi and ii the quotient ring r i g where i g m1 mq they come together in the construction of the rees algebra r i g of the ideal i g the emphasis is almost entirely on sets of squarefree monomials of degree two and their attached graphs the main results are assertions about the cohen macaulay behaviour of the koszul homology of i g and how normality or cohen macaulayness of one of the algebras can be read off the properties of the graph or in the other algebra

a blow up result for dyadic models of the euler equations we partially answer a question raised by kiselev and zlatos int math res not 2005 38 2315 2339 2005 in the generalized dyadic model of the euler equation a blow up of h 1 3 delta norm occurs we recover a few previous blow up results for various related dyadic models as corollaries

block diagonally preconditioned piu methods of saddle point problem for large sparse saddle point problems we firstly introduce the block diagonally preconditioned gauss seidl method pbgs which reduces to the gsor method z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 and piu method z z bai z q wang on parameterized inexact uzawa methods for generalized saddle point problems linear algebra appl 428 2008 2900 2932 when the preconditioners equal to different matrices respectively then we generalize the pbgs method to the ppiu method and discuss the sufficient conditions such that the spectral radius of the ppiu method is much less than one furthermore some rules are considered for choices of the preconditioners including the splitting method of the 1 1 block matrix in the piu method and numerical examples are given to show the superiority of the new method to the piu method

face rings of simplicial complexes with singularities the face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular i e has the homology of a wedge of spheres of the expected dimension this is derived from an enumerative result for local cohomology of face rings modulo generic linear forms as compared with local cohomology of the face ring itself the enumerative result is generalized to squarefree modules a concept of cohen macaulay in codimension c is defined and characterized for arbitrary finitely generated modules and coherent sheaves for the face ring of an r dimensional complex it is equivalent to nonsingularity of in dimension r c for a coherent sheaf on projective space this condition is shown to be equivalent to the same condition on any single generic hyperplane section the characterization of nonsingularity in dimension m via finite local cohomology thus generalizes from face rings to arbitrary graded modules

a classification of the face numbers of buchsbaum simplicial posets the family of buchsbaum simplicial posets generalizes the family of simplicial cell manifolds the h vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced betti numbers of its geometric realization novik and swartz showed that the h vector of a buchsbaum simplicial poset satisfies certain simple inequalities in this paper we show that these necessary conditions are in fact sufficient to characterize the h vectors of buchsbaum simplicial posets with prescribed betti numbers

depth and regularity of powers of sums of ideals given arbitrary homogeneous ideals i and j in polynomial rings a and b over a field k we investigate the depth and the castelnuovo mumford regularity of powers of the sum i j in a otimes k b in terms of those of i and j our results can be used to study the behavior of the depth and regularity functions of powers of an ideal for instance we show that such a depth function can take as its values any infinite non increasing sequence of non negative integers

buchsbaumness of ordinary powers of two dimensional square free monomial ideals abstract let s k x 1 x 2 x n be a polynomial ring let i be a stanley reisner ideal in s of a pure simplicial complex of dimension one in this paper we study the buchsbaum property of s i r for any integer r 0 our first purpose is giving a characterization of ext modules ext s p s m t s j for any monomial ideal j where m t x 1 t x 2 t x n t in terms of certain simplicial complexes then we consider the buchsbaum property of s i r the main tool to check the buchsbaumness is the surjectivity criterion we see the behavior of the canonical map from ext s p s m t s i r to h m p s i r from the view point of reduced cohomology groups of simplicial complexes

applications of dualizing complexes to buchsbaum rings let a m k denote a local noetherian ring with d dim a admitting a normalized dualizing complex d by well known theorems on local duality a is cohen macaulay ring if and only if the d th place truncated dualizing complex r d is isomorphic to the zero complex in the derived category d a see section 1 for more precise statements in the following we consider a class of local rings for which the truncated dualizing complex t d is isomorphic to a complex of vector spaces over a m k there have been several efforts to get results about local rings which generalize the cohen macaulay property see for instance 7 13 14 a connection to our consideration is given in the following sense in 2 3 we prove one of our main theorems the complex t d of a is isomorphic to a complex of vector spaces over k if and only if a is buchsbaum ring in the sense of 13 141 they are those rings for which the difference l a d e x a between length and multiplicity of a system of parameters x does ndt depend on x see also 2 1 for a brief summary of essential facts about buchsbaum rings and relation to a conjecture of d a buchsbaum in 7 m hochster and j l roberts have studied purity of the frobenius map and related things to show that the local cohomology modules of certain rings of such type behave nicely applying our criterion 3 1 to this situation we can show in 4 1 that such rings have a truncated dualizing complex t d which is isomorphic to a complex of vector spaces over k following the examples given in 7 we get a number of buchsbaum rings for instance let vc ip be a projective variety defined by an ideal which is generated by square free monomials then v is arithmetically buchsbaum if and only if v is a cohen macaulay variety our theorem 2 3 is related to a result of r kiehl in 8 using the surjective map criterion of 141 he showed that the canonical module of a is buchsbaum if t d is isomorphic to a complex of vector spaces over k this yields in 8 that the local rings at the cusps of hilbert modular groups are factorial buchsbaum rings which are not cohen macaulay 61 oool 8708 82 040061 17 05 00 o

symbolic powers of sums of ideals let i and j be nonzero ideals in two noetherian algebras a and b over a field k let i j denote the ideal generated by i and j in a otimes k b we prove the following expansion for the symbolic powers begin aligned i j n sum i j n i i j j end aligned if a and b are polynomial rings and if mathrm char k 0 or if i and j are monomial ideals we give exact formulas for the depth and the castelnuovo mumford regularity of i j n which depend on the interplay between the symbolic powers of i and j the proof involves a result of independent interest which states that the induced map mathrm tor i a k i n rightarrow mathrm tor i r k i n 1 is zero for any homogeneous ideal i and i ge 0 n ge 0 we also investigate other properties and invariants of i j n such as the equality between ordinary and symbolic powers the waldschmidt constant and the cohen macaulayness

matroid configurations and symbolic powers of their ideals star configurations are certain unions of linear subspaces of projective space that have been studied extensively we develop a framework for studying a substantial gen eralization which we call matroid configurations whose ideals generalize stanley reisner ideals of matroids such a matroid configuration is a union of complete intersections of a fixed codimension relating these to the stanley reisner ideals of matroids and using methods of liaison theory allows us in particular to describe the hilbert function and minimal generators of the ideal of what we call a hypersurface configuration we also establish that the symbolic powers of the ideal of any matroid configuration are cohen macaulay as applications we study ideals coming from certain complete hypergraphs and ideals derived from tetrahedral curves we also consider waldschmidt constants and resurgences in particular we determine the resurgence of any star configuration and many hypersurface configurations previously the only non trivial cases for which the resurgence was known were certain monomial ideals and ideals of finite sets of points finally we point out a connection to secant varieties of varieties of reducible forms

a generalized preconditioned parameterized inexact uzawa method for singular saddle point problems in this short paper we introduce and analyze a generalized preconditioned parameterized inexact uzawa method for solving singular saddle point problems this method is a generalization of the ppiu method see gao and kong 2010 then we give the semi convergent conditions of this method which are more practical for the applications

spectral properties of the matrix splitting preconditioners for generalized saddle point problems abstract in this paper we study the spectral properties of two different kinds of preconditioners for generalized saddle point problems one is based on the two parameter matrix splitting preconditioner for saddle point problems by wang et al we generalize this preconditioner to generalized saddle point problems and analyze the spectral properties of the corresponding preconditioned matrix the other is based on the hermitian and skew hermitian splitting hss preconditioner for generalized saddle point problems by huang et al we study the spectral properties of the hss preconditioner with two different parameters and the generalized hss preconditioner for generalized saddle point problems in addition some numerical tests are given to verify the validity of the presented theoretical results

cohen macaulayness of monomial ideals and symbolic powers of stanley reisner ideals abstract we present criteria for the cohen macaulayness of a monomial ideal in terms of its primary decomposition these criteria allow us to use tools of graph theory and of linear programming to study the cohen macaulayness of monomial ideals which are intersections of prime ideal powers we can characterize the cohen macaulayness of the second symbolic power or of all symbolic powers of a stanley reisner ideal in terms of the simplicial complex these characterizations show that the simplicial complex must be very compact if some symbolic power is cohen macaulay in particular all symbolic powers are cohen macaulay if and only if the simplicial complex is a matroid complex we also prove that the cohen macaulayness can pass from a symbolic power to another symbolic powers in different ways

equality of ordinary and symbolic powers of stanley reisner ideals abstract this paper studies properties of simplicial complexes with the equality i m i m for a given m 2 the main results are combinatorial characterizations of such complexes in the two dimensional case it turns out that there exists only a finite number of complexes with this property and that these complexes can be described completely as a consequence we are able to determine all complexes for which i m is cohen macaulay for some m 2 in particular there are complexes with i 2 i 2 or i 3 i 3 but i m i m for all m 4 and that if i m i m for some m 4 then i m i m for all m 1 similarly there are complexes for which i 2 is cohen macaulay but i m is not cohen macaulay for all m 3 and if i m is cohen macaulay for some m 3 then i is a complete intersection

perfect linkage of cohen macaulay modules over cohen macaulay rings abstract in this paper we introduce and study the notion of linkage by perfect modules which we call perfect linkage for cohen macaulay modules over cohen macaulay local rings we explore perfect linkage in connection with syzygies maximal cohen macaulay approximations and yoshino isogawa linkage we recover a theorem of yoshino and isogawa and analyze the structure of double perfect linkage moreover we establish a criterion for two cohen macaulay modules of codimension one to be perfectly linked and apply it to the classical linkage theory for ideals we also construct various examples of linkage of modules and ideals

ext and local cohomology modules of face rings of simplicial posets abstract there are a large number of theorems detailing the homological properties of the stanley reisner ring of a simplicial complex here we attempt to generalize some of these results to the case of a simplicial poset by investigating the combinatorics of certain modules associated with the face ring of a simplicial poset from a topological viewpoint we extend some results of miyazaki and grabe to a wider setting

depth functions of symbolic powers of homogeneous ideals this paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal our investigation is focused on the behavior of the function mathrm depth r i t dim r mathrm pd i t 1 where i t denotes the t th symbolic power of a homogeneous ideal i in a noetherian polynomial ring r and mathrm pd denotes the projective dimension it has been an open question whether the function mathrm depth r i t is non increasing if i is a squarefree monomial ideal we show that mathrm depth r i t is almost non increasing in the sense that mathrm depth r i s ge mathrm depth r i t for all s ge 1 and t in e s where begin aligned e s bigcup i ge 1 t in mathbb n i s 1 1 le t le is end aligned which contains all integers t ge s 1 2 1 the range e s is the best possible since we can find squarefree monomial ideals i such that mathrm depth r i s mathrm depth r i t for t not in e s which gives a negative answer to the above question another open question asks whether the function mathrm depth r i t is always constant for t gg 0 we are able to construct counter examples to this question by monomial ideals on the other hand we show that if i is a monomial ideal such that i t is integrally closed for t gg 0 e g if i is a squarefree monomial ideal then mathrm depth r i t is constant for t gg 0 with begin aligned lim t rightarrow infty mathrm depth r i t dim r dim oplus t ge 0 i t mathfrak m i t end aligned our last result which is the main contribution of this paper shows that for any positive numerical function phi t which is periodic for t gg 0 there exist a polynomial ring r and a homogeneous ideal i such that mathrm depth r i t phi t for all t ge 1 as a consequence for any non negative numerical function psi t which is periodic for t gg 0 there is a homogeneous ideal i and a number c such that mathrm pd i t psi t c for all t ge 1

deformation and linkage of gorenstein algebras general double linkage of gorenstein algebras is defined rigidity genericity and regularity up to codimension six all pass across general double linkage rigid strongly unobstructed codimension four gorenstein algebras which lie in different herzog classes are produced introduction we begin by reviewing the classical structure theorems let p k xi x 41 a p i and f be the minimal resolution of a by free p modules f 1 if a has codimension 1 and i is unmixed then f is given by 0 p p for some fin p 2 hilbert burch 7 schaps 28 if a has codimension 2 and is cohen macaulay then f is given by x x 0 p n i ppnp where xis an n x n 1 matrix xi l 1l i x and a i x is the determinant formed by omitting the ith row from x 3 buchsbaum and eisenbud 4 if a has codimension 3 and its gorenstein then f is given by xt x x o p pn pn p where xis an n x n alternating matrix xi l i lpfi x and pfi x denotes the pfaffian of the alternating submatrix of x formed by omitting row and column i in these structure theorems as the codimension of a increases so does the number of conditions on a it is reasonable to suspect that a structure theorem for codimension four gorenstein algebras a will require that an additional hypothesis be imposed on a we observe that in each case a there is a structure theorem b all deformations are unobstructed see herzog 13 and c each algebra a is in the linkage class of a complete intersection see peskine and szpiro 27 and watanabe 30 historically these three facts were found independently with hindsight we now see that b follows from c and a follows in turn from b in his thesis received by the editors june 10 1982 and in revised form march 29 1983 some of the contents of this paper were presented to the special session on commutative algebra at the american mathematical society meeting in cincinnati on january 13 1982 1980 mathematics subject classification primary 13c05 13d25 13h10 secondary 14b07 14d115 14m05 16ao3 both authors were supported in part by the national science foundation 1984 american mathematical society 0002 9947 84 1 00 25 per page

locality and stability of the cascades of two dimensional turbulence we investigate and clarify the notion of locality as it pertains to the cascades of two dimensional turbulence the mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions first introduced by l vov and procaccia as a point of departure we use a revised version of the system of hypotheses that was proposed by frisch for three dimensional turbulence we show that both the enstrophy cascade and the inverse energy cascade are local in the sense of non perturbative statistical locality we also investigate the stability conditions for both cascades we have shown that statistical stability with respect to forcing applies unconditionally for the inverse energy cascade for the enstrophy cascade statistical stability requires large scale dissipation and a vanishing downscale energy dissipation a careful discussion of the subtle notion of locality is given at the end of the paper

on asymptotic vanishing behavior of local cohomology let r be a standard graded algebra over a field k with irrelevant maximal ideal mathfrak m and i a homogeneous r ideal we study the asymptotic vanishing behavior of the graded components of the local cohomology modules text h i mathfrak m r i n n in mathbb n for i dim r i we show that when mathrm char k 0 r i is cohen macaulay and i is a complete intersection locally on mathrm spec r setminus mathfrak m the lowest degrees of the modules text h i mathfrak m r i n n in mathbb n are bounded by a linear function whose slope is controlled by the generating degrees of the dual of i i 2 our result is a direct consequence of a related bound for symmetric powers of locally free modules if no assumptions are made on the ideal or the field k we show that the complexity of the sequence of lowest degrees is at most polynomial provided they are finite our methods also provide a result on stabilization of maps between local cohomology of consecutive powers of ideals

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

locality of turbulent cascades in this paper we establish sufficient conditions for locality of turbulent cascades by an exact analysis of the fluid equations no assumptions of homogeneity or isotropy are invoked and the results apply to individual realizations with no resort to any statistical averaging the only requirement is suitable regularity of the turbulent solutions of the fluid equations in the high reynolds number limit corresponding to holder continuity but non differentiability in space we use a smooth filtering approach to resolve the turbulent fields both in space and in scale we discuss several physical cascades to exemplify and clarify the analysis including joint energy and helicity cascades in three space dimensions and the dual inverse energy cascade and direct enstrophy cascade in two dimensions

cohen macaulayness of large powers of stanley reisner ideals abstract we prove that for m 3 the symbolic power i m of a stanley reisner ideal is cohen macaulay if and only if the simplicial complex is a matroid similarly the ordinary power i m is cohen macaulay for some m 3 if and only if i is a complete intersection these results solve several open questions on the cohen macaulayness of ordinary and symbolic powers of stanley reisner ideals moreover they have interesting consequences on the cohen macaulayness of symbolic powers of facet ideals and cover ideals

onsager s conjecture on the energy conservation for solutions of euler s equation we give a simple proof of a result conjectured by onsager 1 on energy conservation for weak solutions of euler s equation

on the cohen macaulayness of the conormal module of an ideal abstract in the present paper we investigate a question stemming from a long standing conjecture of vasconcelos given a generically complete intersection perfect ideal i in a regular local ring r when is it true that the cohen macaulayness of i i 2 or r i 2 implies that r i is gorenstein this property is known to hold for licci ideals and essentially squarefree monomial ideals we show that a positive answer actually holds for every monomial ideal we then give a positive answer for several special classes of ideals and provide application to algebroid curves with low multiplicity we also exhibit prime ideals in regular local rings and homogeneous level ideals in polynomial rings for which the answer is negative and use them to show the sharpness of our main result as they lie in the first class of ideals not covered by it the homogeneous examples have been found thanks to the help of j c migliore as a by product we exhibit several classes of cohen macaulay ideals whose square is not cohen macaulay our methods work both in the homogeneous and in the local settings

on semi convergence of parameterized uzawa methods for singular saddle point problems abstract bai et al recently proposed an efficient parameterized uzawa method for solving the nonsingular saddle point problems z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 in this paper we further prove the semi convergence of this method when it is applied to solve the singular saddle point problems under suitable restrictions on the involved iteration parameters the optimal iteration parameters and the corresponding optimal semi convergence factor for the parameterized uzawa method are determined in addition numerical experiments are used to show the feasibility and effectiveness of the parameterized uzawa method for solving singular saddle point problems

the thermodynamics of fractals revisited with wavelets abstract the multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform this new approach is based on the definition of partition functions from the wavelet transform modulus maxima we demonstrate that very much like thermodynamic functions the generalized fractal dimensions d q and the f singularity spectrum can be readily determined from the scaling behavior of these partition functions we show that this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized boxes we illustrate our theoretical considerations on pedagogical examples e g devil s staircases and fractional brownian motions we also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data that we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects in particular we show that a dynamical system which leaves invariant such an object can be uncovered form the space scale arrangement of its wavelet transform modulus maxima we elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem this step towards a statistical mechanics of fractals is illustrated on discrete period doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos finally we apply our technique to analyze the fractal hierarchy of dla azimuthal cantor sets defined by intersecting the inner frozen region of large mass off lattice diffusion limited aggregates dla wit a circle this study clearly lets out the existence of an underlying multiplicative process that is likely to account for the fibonacci structural ordering recently discovered in the apparently disordered arborescent dla morphology

a new sor like method for the saddle point problems abstract in previous years golub wu and yuan presented a generalized successive over relaxation sor like method for solving the saddle point problems in this paper we present a new sor like nsor like method which has three parameters our new method can be applied to the nonsingular saddle point problems as well as the singular cases the characteristic of eigenvalues of the iteration matrix of this nsor like method is analyzed then we give the convergence semi convergence theorem of the new iterative method by giving the restrictions imposed on the parameter moreover that convergence semi convergence theorem is applied to some special cases to give the convergence region for the parameters we can see that nsor like method has a wider convergence semi convergence region for and than the parameterized uzawa method which covers preconditioned uzawa method and uzawa method in addition the optimal iteration parameters and the corresponding convergence semi convergence factor for the uzawa method are presented

universal direct cascade in two dimensional turbulence we consider the correlation functions of vorticity ensuremath omega in the region of the direct cascade in a steady two dimensional turbulence the nonlocality of the cascade in k space provides for logarithmic corrections to the expressions obtained by dimension estimates and the main problem is to take those logarithms into account our procedure starts directly with the euler equation rewritten in the comoving reference frame we express the correlation functions of the vorticity via the correlation functions of the pumping force and renormalized strain it enables us to establish a set of integrodifferential equations which gives a logarithmic renormalization of the vorticity correlation functions in the inertial interval we find the indices characterizing the logarithmic behavior of different correlation functions for example the two point simultaneous functions are as follows mathrm ensuremath omega mathit n mathbf r 1 mathrm ensuremath omega mathit n mathbf r 2 ensuremath sim mathit p 2 ln l mathbf r 1 mathbf r 2 ensuremath vert 2 mathit n 3 where l is the pumping scale we demonstrate that the form of those correlation functions is universal i e independent of the pumping the only pumping related value which enters the expressions is the enstrophy production rate mathit p 2 the contributions related to pumping rates mathit p mathit n of the higher order integrals of motion are demonstrated to be small in comparison with the ones induced by mathit p 2 we establish also the time dependence of the correlation functions the correlation time ensuremath tau in the comoving reference frame is the same for the vorticity and strain and is scale dependent ensuremath tau ensuremath propto mathrm ln 2 3 l r we reformulate our procedure in the diagrammatic language to reinforce the conclusions

restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems the restrictively preconditioned conjugate gradient rpcg method for solving large sparse system of linear equations of a symmetric positive definite and block two by two coefficient matrix is further studied in fact this rpcg method is essentially the classical preconditioned conjugate gradient pcg method with a specially structured preconditioner within this setting we present algorithmic descriptions of two restrictive preconditioners that respectively employ the block jacobi and the block symmetric gauss seidel matrix splitting matrices as approximations to certain matrices involved in them and give convergence analyses of the correspondingly induced two pcg methods numerical results show that these restrictive preconditioners can lead to practical and effective pcg methods for solving large sparse systems of linear equations of symmetric positive definite and block two by two coefficient matrices

rigidity and the lower bound theorem 1 for an arbitrary triangulated d 1 manifold without boundaryc withf 0 vertices andf 1 edges define gamma c f 1 df 0 left begin array 20 c d 1 2 end array right barnette proved that c 0 we use the rigidity theory of frameworks and in particular results related to cauchy s rigidity theorem for polytopes to give another proof for this result we prove that ford 4 if c 0 thenc is a triangulated sphere and is isomorphic to the boundary complex of a stacked polytope other results a we prove a lower bound conjectured by bjorner for the number ofk faces of a triangulated d 1 manifold with specified numbers of interior vertices and boundary vertices b ifc is a simply connected triangulatedd manifold d 4 and lk v c 0 for every vertexv ofc then c 0 lk v c is the link ofv inc c letc be a triangulatedd manifold d 3 then ske11 d 2 can be embedded in skel1 c iff c 0 d is thed dimensional simplex d ifp is a 2 simpliciald polytope then f 1 p geqq df 0 p left begin array 20 c d 1 2 end array right related problems concerning pseudomanifolds manifolds with boundary and polyhedral manifolds are discussed

local energy flux and the refined similarity hypothesis in this paper we demonstrate the locality of energy transport for incompressible euler equations both in space and in scale the key to the proof is the proper definition of a local subscale flux t r which is supposed to be a measure of energy transfer to length scales l at the space point r kraichnan suggested that for such a quantity the refined similarity hypothesis will hold which kolmogorov originally assumed to hold instead for volume averaged dissipation we derive a local energy balance relation for the large scale motions which yields a natural definition of such a subscale flux for this definition a precise form of the refined similarity hypothesis is rigorously proved as a big o bound the established estimate is t r o l3h 1 in terms of the local holder exponenth at the point r which is also the estimate assumed in the parisi frisch multifractal model our method not only establishes locality of energy transfer but it also clarifies the physical reason that convection effects which naively violate locality do not contribute to the subscale flux in fact we show that as a consequence of incompressibility such effects enter into the local energy balance only as the divergence of a spatial current therefore they describe motion of energy in space and cancel in the integration over volume we also discuss theorems of onsager eyink and constantinet al on energy conservation for euler dynamics particularly to explain their relation with the parisi frisch model the constantinet al proof may be interpreted as giving a bound on the total flux t d d r t r of the form t r o lz3h 1 wherez3 is the third order scaling exponent or besov index in agreement with the multifractal model finally we discuss how the local estimates are related to the results of caffarelli kohn nirenberg on partial regularity for solutions of navier stokes equations they provide some heuristic support to a scenario proposed recently by pumir and siggia for singularities in the solutions of navier stokes with small enough viscosity

exact results on stationary turbulence in 2d consequences of vorticity conservation we establish a series of exact results for a model of stationary turbulence in two dimensions the incompressible navier stokes equation in a periodic domain with stochastic force white noise in time essentially all of our conclusions follow from the simple consideration of the simultaneous conservation of energy and enstrophy by the inertial dynamics our main results are as follows 1 we show the blow up of mean energy as l02e for 0 when there is no ir dissipation at the large length scale l0 2 with an additional ir dissipation we establish the validity of the traditional cascade directions and magnitudes of flux of energy and enstrophy for 0 assuming finite mean energy in the limit 3 we rigorously establish the balance equations for the energy and vorticity invariants in the 2d steady state and the forward cascade of the higher order vorticity invariants assuming finite mean values 4 we derive exact inequalities for scaling exponents in the 2d enstrophy range as follows if l p l p then 2 23 and p 0 for p 3 if the minimum holder exponent of the vorticity hmin 0 then we establish a 2d analogue of the refined similarity hypothesis which improves these bounds the most novel and interesting conclusion of this work is the connection established between intermittency in 2d and negative holder singularities of the vorticity we show that the latter are necessary for deviations from the 1967 kraichnan scaling to occur

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

wavelet transforms of self similar processes this paper deals with the wavelet transforms of self similar random processes of the kind assumed in the kolmogorov 1941 theory of turbulence it is shown that after suitable rescaling the wavelet transform at a given position becomes a stationary random function of the logarithm of the scale argument in the transform the rescaling depends on the scaling exponent the statistical properties of the resulting random fluctuations such as its correlation function depend on the choice of the analyzing wavelet some implications are i the presence of fluctuations in log log plots of the absolute value of wavelet transforms versus the scale ii an estimate of the small scale fluctuations generalizing the iterated logarithm law of brownian motion iii an ultraviolet ergodic formula giving the scaling exponents in terms of zoom averages some observations are made on the issue of fluctuations observed in wavelet transforms of turbulence data

constraint preconditioners for solving singular saddle point problems we study constraint preconditioners for solving singular saddle point problems we analyze properties of the preconditioned matrices in particular their eigenvalue distributions and prove that for solving singular saddle point problems by preconditioned gmres methods with constraint preconditioners gmres will determine the least squares solutions at breakdown in addition we present some numerical examples to demonstrate the convergence behavior of preconditioned gmres for solving singular saddle point problems

a generalized lower bound theorem for balanced manifolds a simplicial complex of dimension d 1 is said to be balanced if its graph is d colorable juhnke kubitzke and murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes we establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orientable homology manifolds with boundary under an additional assumption that all proper links of these triangulations have the weak lefschetz property as a corollary we show that if delta is an arbitrary balanced triangulation of any closed homology manifold of dimension d 1 ge 3 then 2h 2 delta d 1 h 1 delta ge 4 d atopwithdelims 2 tilde beta 1 delta tilde beta 0 delta thus verifying a conjecture by klee and novik to prove these results we develop the theory of flag h vectors

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

saturation and associated primes of powers of edge ideals abstract for the edge ideal i of an arbitrary simple graph we describe the monomials of the saturation of i t in terms of vertex weighted graphs associated with the monomials this description allows us to characterize the embedded associated primes of i t as covers of which contain certain types of subgraphs of as an application we completely classify the associated primes of i 2 and i 3 in terms of

the basic equations for the large eddy simulation of turbulent flows in complex geometry abstract the equations for large eddy simulation les of turbulent flows are derived by applying a filtering operation to the navier stokes equations les of inhomogeneous turbulent flows requires the use of filters with variable filter width the use of such filters invalidates the standard derivation of the basic equations for the filtered fields since the filtering operation in general does not commute with the operation of differentiation in this paper an alternate definition of the filtering operation based on the mapping function of the nonuniform grid is introduced it is shown that with this modified definition the filtering and differentiation operations commute up to an error which is second order in the filter width it is also shown that the commutation error can be expressed in terms of the filtered field and its derivatives as an asymptotic series in the square of the filter width these results are then applied to the navier stokes equations to derive the basic equations satisfied by the filtered fields

linkage and the koszul homology of ideals 0 introduction let r be a local ring and i an ideal of r associated to i are several graded algebras the symmetric algebra of the module i sym i the rees algebra of i defined to be the algebra r i gn o inf and the associated graded algebra of i gr1 r rii g ii 2 in in 1 i i i our purpose in this paper is to study the behavior of these graded algebras when we change i through linkage recall the definition of linkage found in the paper of peskine and szpiro 22

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

besov spaces and the multifractal hypothesis parisi and frisch proposed some time ago an explanation for multiscaling of turbulent velocity structure functions in terms of a multifractal hypothesis i e they conjecture that the velocity field has local holder exponents in a range hmin hmax with exponents h occurring on a sets h with a fractal dimensiond h heuristic reasoning led them to an expression for the scaling exponentz p ofpth order as the legendre transform of the codimensiond d h we show here that a part of the multifractal hypothesis is correct under even weaker assumptions namely if the velocity field hasl p mean holder indexs i e if it lies in the besov spaceb p s then local holder regularity is satisfied ifs d p then the hypothesis is true in a generalized sense of holder space with negative exponents and we discuss the proper definition of local holder classes of negative index finally if a certain box counting dimension exists then the legendre transform of its codimension gives the scaling exponentz p and more generally the maximal besov index of order p ass p z p p our method of proof is derived from a recent paper of s jaffard using compactly supported orthonormal wavelet bases and gives an extension of his results we discuss implications of the theorems for ensemble average scaling and fluid turbulence

symbolic powers of monomial ideals and vertex cover algebras abstract we introduce and study vertex cover algebras of weighted simplicial complexes these algebras are special classes of symbolic rees algebras we show that symbolic rees algebras of monomial ideals are finitely generated and that such an algebra is normal and cohen macaulay if the monomial ideal is squarefree for a simple graph the vertex cover algebra is generated by elements of degree 2 and it is standard graded if and only if the graph is bipartite we also give a general upper bound for the maximal degree of the generators of vertex cover algebras

multilocality and fusion rules on the generalized structure functions in two dimensional and three dimensional navier stokes turbulence using the fusion rules hypothesis for three dimensional and two dimensional navier stokes turbulence we generalize a previous nonperturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences we call this generalization of nonperturbative locality to multiple applications of the nonlinear interactions operator multilocality the resulting cross terms pose a new challenge requiring a new argument and the introduction of a new fusion rule that takes advantage of rotational symmetry our main result is that the fusion rules hypothesis implies both locality and multilocality in both the ir and uv limits for the downscale energy cascade of three dimensional navier stokes turbulence and the downscale enstrophy cascade and inverse energy cascade of two dimensional navier stokes turbulence we stress that these claims relate to nonperturbative locality of generalized structure functions on all orders and not the term by term perturbative locality of diagrammatic theories or closure models that involve only two point correlation and response functions

level property of ordinary and symbolic powers of stanley reisner ideals abstract in this paper we prove that the t th ordinary and or symbolic power of a stanley reisner ideal is level for some positive integer t 3 if and only if i is a complete intersection and equi generated for t 2 we give a characterization of level property of the second symbolic power i 2 when is a matroid complex of dimension one

convergence properties of preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite matrices for the non hermitian and positive semidefinite systems of linear equations we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned hermitian and skew hermitian splitting iteration methods we then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned hermitian and skew hermitian splitting iteration methods

on the semi convergence of regularized hss iteration methods for singular saddle point problems abstract recently bai and benzi proposed a class of regularized hermitian and skew hermitian splitting rhss iteration methods for solving the nonsingular saddle point problem in this paper we apply this method to solve the singular saddle point problem in the process of the semi convergence analysis we get that the rhss method and the hss method are unconditionally semi convergent which has improved the previous results then some spectral properties of the corresponding preconditioned matrices and a class of improved preconditioned matrices are analyzed finally some numerical experiments on linear systems arising from the discretization of the stokes equation are presented to illustrate the feasibility and effectiveness of this method and the corresponding preconditioners

inverse energy cascade in two dimensional turbulence deviations from gaussian behavior high resolution numerical simulations of stationary inverse energy cascade in two dimensional turbulence are presented deviations from gaussian behavior of velocity differences statistics are quantitatively investigated the level of statistical convergence is pushed enough to permit reliable measurement of the asymmetries in the probability distribution functions of longitudinal increments and odd order moments which bring the signature of the inverse energy flux no measurable intermittency corrections could be found in their scaling laws the seventh order skewness increases by almost two orders of magnitude with respect to the third thus becoming of order unity

standard graded vertex cover algebras cycles and leaves the aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras this property has several nice consequences for the squarefree monomial ideals defining these algebras it turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees these relationships allow us to obtain very general results on standard graded vertex cover algebras which cover previous major results on rees algebras of squarefree monomial ideals

asymptotic behavior of integer programming and the stability of the castelnuovo mumford regularity the paper provides a connection between commutative algebra and integer programming and contains two parts the first one is devoted to the asymptotic behavior of integer programs with a fixed cost linear functional and the constraint sets consisting of a finite system of linear equations or inequalities with integer coefficients depending linearly on n an integer n is determined such that the optima of these integer programs are a quasi linear function of n for all n ge n using results in the first part one can bound in the second part the indices of stability of the castelnuovo mumford regularities of integral closures of powers of a monomial ideal and that of symbolic powers of a square free monomial ideal

licci squarefree monomial ideals generated in degree two or with deviation two abstract we study squarefree monomial ideals which are in the linkage class of a complete intersection we call such an ideal licci firstly we classify all licci edge ideals secondly we prove that any cohen macaulay almost complete intersection squarefree monomial ideal is licci thirdly we characterize licci squarefree monomial ideals of deviation 2 in terms of hypergraphs associated to the ideals as an application of the classification in kimura terai and yoshida 2009 15

residual intersections in cohen macaulay rings abstract residual intersection generalizes the notion of linkage the central questions in residual intersection theory are for a residual intersection j of i when is r j cohen macaulay and what is the canonical module for r j our main result gives answers to the two central questions in the case where r is merely cohen macaulay

stability of depths of powers of edge ideals abstract let g be a graph and let i i g be its edge ideal in this paper we provide an upper bound of n from which depth r i g n is stationary and compute this limit explicitly this bound is always achieved if g has no cycles of length 4 and every its connected component is either a tree or a unicyclic graph

linkage theory for algebras with pure resolutions abstract let s be a graded cohen macaulay quotient r i of a polynomial ring r k x 1 x n over an infinite field k if s is in the linkage class of a complete intersection then its koszul homology and twisted conormal module are cohen macaulay by passing to zero and one dimensional specializations of s one can convert these properties into numerical test conditions for a graded algebra to be in the linkage class of a complete intersection we prove that if s is generically a complete intersection and the minimal free solution of s is pure and almost linear then s is not in the linkage class of a complete intersection apart from a few obvious exceptions for algebras of small codimension the results are stronger if s has codimension three and has a pure resolution then s is in the linkage class of a complete intersection if and only if s is gorenstein if s is a codimension four gorenstein algebra with a pure resolution then s is in the linkage class of a complete intersection if and only if s is a complete intersection

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

semi convergence analysis of uzawa methods for singular saddle point problems recently zheng bai and yang studied the parameterized uzawa method for solving singular saddle point problems b zheng z z bai x yang on semi convergence of parameterized uzawa methods for singular saddle point problems linear algebra appl 431 2009 808 817 in this paper we discuss the inexact uzawa method which covers the uzawa method the preconditioned uzawa method and the parameterized uzawa method to solve the singular saddle point problems we prove the semi convergence result under restrictions by verifying two necessary and sufficient conditions that is all elementary divisors associated with the eigenvalue 1 of its iterative matrix are linear and the pseudo spectral radius of the iterative matrix is less than 1 sufficient conditions for the semi convergence of several uzawa type methods are also provided in addition numerical examples are given to demonstrate the semi convergence of uzawa type methods

single scale wavelet representation of turbulence dynamics formulation and navier stokes regularity abstract the triad interactions arising from the spectral resolution of the nonlinear terms in the navier stokes equations have so far not been substantially modified in the wavelet representation in this paper the multiscale interactions are captured by exact expressions evaluated at a single scale of the mexican hat wavelet coefficients the larger scale terms as a volume integral of nearby wavelet coefficients and the smaller scale contributions as iterated laplacians of the coefficient at the point of interest as a result the navier stokes equations are expressed exactly at a single scale this facilitates the evaluation of the dominant holder exponent near singularities from the scaling properties of wavelet coefficients it is shown that euler dynamics would generate stronger singularities for any h 1 but that viscous dynamics would not unless h 1 a discontinuous case we discuss how this conclusion could be affected by boundary conditions

a generalization of parameterized inexact uzawa method for singular saddle point problems in this paper we further study the generalized parameterized inexact uzawa method for solving singular saddle point problems obtaining the generalized parameterized inexact uzawa gpiu method theoretical analysis shows that the semi convergence of this new method can be guaranteed by suitable choices of the iteration parameters numerical experiments are used to demonstrate the feasibility and effectiveness of the generalized parameterized inexact uzawa method for solving singular saddle point problems

matrix exponential based closures for the turbulent subgrid scale stress tensor two approaches for closing the turbulence subgrid scale stress tensor in terms of matrix exponentials are introduced and compared the first approach is based on a formal solution of the stress transport equation in which the production terms can be integrated exactly in terms of matrix exponentials this formal solution of the subgrid scale stress transport equation is shown to be useful to explore special cases such as the response to constant velocity gradient but neglecting pressure strain correlations and diffusion effects the second approach is based on an eulerian lagrangian change of variables combined with the assumption of isotropy for the conditionally averaged lagrangian velocity gradient tensor and with the recent fluid deformation approximation it is shown that both approaches lead to the same basic closure in which the stress tensor is expressed as the matrix exponential of the resolved velocity gradient tensor multiplied by its transpose short time expansions of the matrix exponentials are shown to provide an eddy viscosity term and particular quadratic terms and thus allow a reinterpretation of traditional eddy viscosity and nonlinear stress closures the basic feasibility of the matrix exponential closure is illustrated by implementing it successfully in large eddy simulation of forced isotropic turbulence the matrix exponential closure employs the drastic approximation of entirely omitting the pressure strain correlation and other nonlinear scrambling terms but unlike eddy viscosity closures the matrix exponential approach provides a simple and local closure that can be derived directly from the stress transport equation with the production term and using physically motivated assumptions about lagrangian decorrelation and upstream isotropy

socles of buchsbaum modules complexes and posets abstract the socle of a graded buchsbaum module is studied and is related to its local cohomology modules this algebraic result is then applied to face enumeration of buchsbaum simplicial complexes and posets in particular new necessary conditions on face numbers and betti numbers of such complexes and posets are established these conditions are used to settle in the affirmative kuhnel s conjecture for the maximum value of the euler characteristic of a 2k dimensional simplicial manifold on n vertices as well as kalai s conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension number of vertices and the first betti number

topological finiteness for edge vertex enumeration the number of pl homeomorphism types of combinatorial manifolds in a fixed dimension with an upper bound on g2 is finite

cohen macaulayness for symbolic power ideals of edge ideals abstract let s k x 1 x n be a polynomial ring over a field k let i g s denote the edge ideal of a graph g we show that the lth symbolic power i g l is a cohen macaulay ideal i e s i g l is cohen macaulay for some integer l 3 if and only if g is a disjoint union of finitely many complete graphs when this is the case all the symbolic powers i g l are cohen macaulay ideals similarly we characterize graphs g for which s i g l has flc as an application we show that an edge ideal i g is complete intersection provided that s i g l is cohen macaulay for some integer l 3 this strengthens the main theorem in crupi et al 2010 3

on face numbers of manifolds with symmetry abstract necessary conditions on the face numbers of cohen macaulay simplicial complexes admitting a proper action of the cyclic group z p z of a prime order are given this result is extended further to necessary conditions on the face numbers and the betti numbers of buchsbaum simplicial complexes with a proper z p z action adin s upper bounds on the face numbers of cohen macaulay complexes with symmetry are shown to hold for all d 1 dimensional buchsbaum complexes with symmetry on n 3 d 2 vertices a generalization of kuhnel s conjecture on the euler characteristic of 2 k dimensional manifolds and sparla s analog of this conjecture for centrally symmetric 2 k manifolds are verified for all 2 k manifolds on n 6 k 3 vertices connections to the upper bound theorem are discussed and its new version for centrally symmetric manifolds is established

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

alexander duality for stanley reisner rings and squarefree nn graded modules abstract let s k x1 xn be a polynomial ring and let s be its canonical module first we will define squarefreeness for n n graded s modules a stanley reisner ring k s i its syzygy module syzi k and extis k s are always squarefree this notion will simplify some standard arguments in the stanley reisner ring theory next we will prove that the i linear strand of the minimal free resolution of a stanley reisner ideal i s has the same information as the module structure of extis k s where is the alexander dual of in particular if k has a linear resolution we can describe its minimal free resolution using the module structure of the canonical module of k which is cohen macaulay in this case we can also give a new interpretation of a result of herzog and co workers which states that k is sequentially cohen macaulay if and only if i is componentwise linear

on block diagonally preconditioned accelerated parameterized inexact uzawa method for singular saddle point problems recently z z bai z q wang on parameterized inexact uzawa methods for generalized saddle point problems linear algebra appl 428 2008 2900 2932 studied a class of parameterized inexact uzawa piu methods and proposed a generalized and modified accelerated parameterized inexact uzawa apiu iteration method for solving nonsingular saddle point problems in this paper we further generalize this method to obtain the block diagonally preconditioned accelerated parameterized inexact uzawa bdp apiu method for solving singular saddle point problems theoretical analysis shows that the semi convergence of this new method can be guaranteed in addition the quasi optimal parameters of the new method are discussed numerical example is given to show the feasibility and effectiveness of the new method for solving singular saddle point problems

regularity of powers of cover ideals of unimodular hypergraphs abstract let h be a unimodular hypergraph over the vertex set n and let j h be the cover ideal of h in the polynomial ring r k x 1 x n we show that reg j h s is a linear function in s for all s r n 2 1 where r is the rank of h moreover for every i a i r j h s is also a linear function in s for s n 2

cohen macaulayness of powers of two dimensional squarefree monomial ideals abstract two dimensional squarefree monomial ideals can be seen as the stanley reisner ideals of graphs the main results of this paper are combinatorial characterizations for the cohen macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph