algebraic shifting and graded betti numbers let s k x 1 x n denote the polynomial ring in n variables over a field k with each deg x i 1 let be a simplicial complex on n 1 n and i c s its stanley reisner ideal we write e for the exterior algebraic shifted complex of a and c for a combinatorial shifted complex of a let ii j i dim k tor i k i i j denote the graded betti numbers of i in the present paper it will be proved that i ii j i e ii j i c for all i and j where the base field is infinite and ii ii j i ii j i c for all i and j where the base field is arbitrary thus in particular one has ii j i ii j i lex for all i and j where lex is the unique lexsegment simplicial complex with the same f vector as a and where the base field is arbitrary

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

maximal minimal resolutions that is exact everywhere except at f0 where the cokernel is m such a resolution always exists if r is a noetherian n graded ring with r0 a field and m is a finitely generated z graded r module then a resolution may be constructed in a minimal way in which each fi is graded and each differential is homogeneous of degree 0 the rank of the ith free module in a minimal free resolution of m is an invariant of m called the ith betti number of m and denoted i m likewise r ij m the number of elements of degree j in a minimal set of homogeneous generators of fi is also an invariant of m called the i j th graded betti number of m in this paper we study modules with maximal graded betti numbers more precisely consider a set consisting of pairs r m where r is a noetherian n graded ring with r0 a field and m is a finitely generated graded r module is there some r m such that ij m ij m for every i and j and every r m there are trivial examples in which the answer is yes e g has only one element or no e g consists of all such pairs theorems 1 and 2 below give affirmative answers for certain sets defined by conditions on the hilbert series and the depths of r and m theorem 3 allows one to compute the maximal graded betti numbers for the pairs in the sets considered in theorems 1 and 2 throughout this paper k denotes a field and q the polynomial ring k x1 xn with the usual n grading given by degxi 1 the hilbert series of a finitely generated graded q module m is hm s dimk mis we define d lexicographic ideals and submodules in section 2 the poincare series of an r module m is p m s t ij m s t we write 4 for coefficient wise inequality of laurent series with coefficients in z

new constructive methods in classical ideal theory abstract the recent development of computer algebra allows us to take up problems of classical ideal theory for which constructive solutions by nearly unfeasible algorithms exist like the problem of deciding whether a polynomial belongs to a given ideal g hermann math ann 95 1926 736 788 even macaulay s methods by using h bases are not satisfactory at least when considering modules the concept of grobner bases introduced in computer algebra by buchberger allows us to find feasible algorithms for many problems of classical ideal theory by transforming them into some low dimensional systems of linear equations our aim is to demonstrate the effectivity of this new concept considering only polynomial rings p k x 1 x n k a field we generalize in part i the notion of grobner bases to modules present some equivalent properties characterizing these bases and show their good computational qualities in part ii we construct starting with grobner bases of ideals explicit free solutions where all the bases of submodules under consideration are also grobner bases and present a reduction strategy for obtaining shorter resolutions where all the bases have also favourable properties t bases part iii deals with applications we propose a method for constructing minimal resolutions starting with the resolutions of part ii and present a convenient way for computing the hilbert function

algebraic invariants for right angled artin groups a finite simplicial graph determines a right angled artin group g with generators corresponding to the vertices of and with a relation w w for each pair of adjacent vertices we compute the lower central series quotients the chen quotients and the first resonance variety of g directly from the graph

monomial golod quotients of exterior algebras it is proved that for any set m of squarefree monomials in the variables x1 xn the algebra a k x1 xn m is golod if and only if the algebra b e x1 xn m is golod where e is the exterior algebra this is proved by showing the equivalence of the extremality of the poincare series of a and b

some wonderful rings in algebraic geometry abstract let a k a1 be a graded ring over a field k the ring is wonderful if torai k k has pure degree i for all i where k a a 1 in a previous paper 4 it was discovered that the ring of regular functions on many cones over abelian varieties is wonderful similarly in 1 backelin has shown that embeddings by high enough trists on ample sheaf on projective schemes give wonderful rings we will estimate high enough and we will note that the coordinate ring of grassmannians in plucker coordinates gives wonderful rings more exactly

applications of mapping cones over clements lindstrom rings abstract we prove that gotzmann s persistence theorem holds over every clements lindstrom ring we also construct the infinite minimal free resolution of a square free borel ideal over such a ring

dynamical analysis of iterative methods for nonlinear systems or how to deal with the dimension abstract this paper deals with the real dynamical analysis of iterative methods for solving nonlinear systems on vectorial quadratic polynomials we use the extended concept of critical point and propose an easy test to determine the stability of fixed points to multivariate rational functions moreover an scaling theorem for different known methods is satisfied we use these tools to analyze the dynamics of the operator associated to known iterative methods on vectorial quadratic polynomials of two variables the dynamical behavior of newton s method is very similar to the obtained in the scalar case but this is not the case for other schemes some procedures of different orders of convergence have been analyzed under this point of view and some dangerous numerical behavior have been found as attracting strange fixed points or periodic orbits

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

generic initial ideals and squeezed spheres in 1988 kalai constructed a large class of simplicial spheres called squeezed spheres and in 1991 presented a conjecture about generic initial ideals of stanley reisner ideals of squeezed spheres in the present paper this conjecture will be proved in order to prove kalai s conjecture based on the fact that every squeezed d 1 sphere is the boundary of a certain d ball called a squeezed d ball generic initial ideals of stanley reisner ideals of squeezed balls will be determined in addition generic initial ideals of exterior face ideals of squeezed balls are determined on the other hand we study the squeezing operation which assigns to each gorenstein complex g having the weak lefschetz property a squeezed sphere sq g and show that this operation increases graded betti numbers

free resolutions of lex ideals over a koszul toric ring in this paper we study the minimal free resolution of lex ideals over a koszul toric ring in particular we study in which toric ring r all lex ideals are componentwise linear we give a certain necessity and sufficiency condition for this property and show that lex ideals in a strongly koszul toric ring are componentwise linear in addition it is shown that in the toric ring arising from the segre product p p every hilbert function of a graded ideal is attained by a lex ideal and that lex ideals have the greatest graded betti numbers among all ideals having the same hilbert function

hilbert schemes and maximal betti numbers over veronese rings macaulay s theorem macaulay in proc lond math soc 26 531 555 1927 characterizes the hilbert functions of graded ideals in a polynomial ring over a field we characterize the hilbert functions of graded ideals in a veronese ring r the coordinate ring of a veronese embedding of pr 1 we also prove that the hilbert scheme which parametrizes all graded ideals in r with a fixed hilbert function is connected this is an analogue of hartshorne s theorem hartshorne in math ihes 29 5 48 1966 that hilbert schemes over a polynomial ring are connected furthermore we prove that each lex ideal in r has the greatest betti numbers among all graded ideals with the same hilbert function

a fast convergent numerical method for matrix sign function with application in sdes in this article a new matrix iterative method for computing the sign of a matrix is offered convergence along with asymptotical stability of the presented method is studied an accelerated form of the iteration will further be constructed finally the application of the given approach in numerical solution of stochastic differential equations and in solving algebraic riccati equations is pointed out

resolutions of a stable ideals abstract let a a 1 a 2 a n be a sequence of integers or we introduce a stable ideals in a polynomial ring and study their homological properties our results generalize results on square free monomial ideals by aramova avramov herzog hibi and srinivasan

hilbert functions of d regular ideals abstract in the present paper we characterize all possible hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to d where d is a positive integer in addition we prove the following result which is a generalization of bigatti hulett and pardue s result let p 0 and d 0 be integers if the base field is a field of characteristic 0 and there is a graded ideal i whose projective dimension proj dim i is smaller than or equal to p and whose regularity reg i is smaller than or equal to d then there exists a monomial ideal l having the maximal graded betti numbers among graded ideals j which have the same hilbert function as i and which satisfy proj dim j p and reg j d we also prove the same fact for squarefree monomial ideals the main methods for proofs are generic initial ideals and combinatorics on strongly stable ideals

a generalized steffensen s method for matrix sign function a direct generalized steffensen s method is proposed for solving the quadratic matrix equation f x x2 i 0 in this way when the matrix a is nonsingular we derive a new numerical scheme for finding matrix sign function the local and global convergence of the proposed generalization is brought up via the concept of basins of attraction we also report some numerical results of the proposed method which show its applicability

periodic solutions of second order nonlinear difference equations with discrete o laplacian we use brouwer degree to prove existence and multiplicity results for the periodic solutions of some nonlinear second order difference equations involving discrete phi laplacian we obtain in particular upper and lower solutions theorems ambrosetti prodi type multiplicity results sharp existence conditions for nonlinearities which are bounded from below or from above and necessary and sufficient conditions for the existence of positive periodic solutions when the nonlinearity is singular at 0 c 2006 elsevier inc all rights reserved

chaotic dynamics of a third order newton type method the dynamics of a classical third order newton type iterative method is studied when it is applied to degrees two and three polynomials the method is free of second derivatives which is the main limitation of the classical third order iterative schemes for systems moreover each iteration consists only in two steps of newton s method having the same derivative with these two properties the scheme becomes a real alternative to the classical newton method affine conjugacy class of the method when is applied to a differentiable function is given chaotic dynamics have been investigated in several examples applying the root finding method to a family of degree three polynomials we have find a bifurcation diagram as those that appear in the bifurcation of the logistic map in the interval

flips and the hilbert scheme over an exterior algebra we study the tangent space at a monomial point m in the hilbert scheme that parameterizes all ideals with the same hilbert function as m over an exterior algebra

a class of quasi newton generalized steffensen methods on banach spaces we consider a class of generalized steffensen iterations procedure for solving nonlinear equations on banach spaces without any derivative we establish the convergence under the kantarovich ostrowski s conditions the majorizing sequence will be a newton s type sequence thus the convergence can have better properties finally a numerical comparation with the classical methods is presented

on a steffensen s type method and its behavior for semismooth equations abstract a generalization of steffensen s method is proposed our goal is to obtain similar convergence as newton s method but without to evaluate any derivative and without to have stability problems convergence analysis and numerical results for semismooth equations are presented