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

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

review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

a generic approach for the solution of nonlinear residual equations part iii sensitivity computations abstract sensitivity analysis provides qualitative and quantitative information on the behaviour of the model under study and offers an access to gradients that may be used for identification purposes such precious information may be obtained at a low development cost applying a generic automatic differentiation ad tool to the computer code implementing this model nonlinear residual problems solved through a path following method may be addressed too in this paper ad techniques are adapted to the taylor based asymptotic numerical method a sensitivity study of a laminated glass beam to the perturbation of some material and geometric parameters and the perturbation of elementary stiffness matrices illustrates the method

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

evolutions symbolic squares and fitting ideals given a reduced local algebra t over a suitable ring or field k we study the question of whether there are nontrivial algebra surjections r t which induce isomorphisms r k t t k such maps called evolutions arise naturally in the study of hecke algebras as they implicitly do in the recent work of wiles taylor wiles and flach we show that the existence of non trivial evolutions of an algebra t can be characterized in terms of the symbolic square of an ideal defining t we give a characterization of the symbolic square in terms of fitting ideals using this and other techniques we show that certain classes of reduced algebras codimension 2 cohen macaulay codimension 3 gorenstein licci algebras in general and some others admit no nontrivial evolutions on the other hand we give examples showing that non trivial evolutions of reduced cohenmacaulay algebras of codimension 3 do exist in every positive characteristic

comparison of symbolic and ordinary powers of ideals all given rings in this paper are commutative associative with identity and noetherian recently l ein r lazarsfeld and k smith els discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0 if h is the largest height of an associated prime of i then i hn i n for all n 0 here if w is the complement of the union of the associated primes of i i t denotes the contraction of i t rw to r where rw is the localization of r at the multiplicative system w their proof depends on the theory of multiplier ideals including an asymptotic version and in particular requires resolution of singularities as well as vanishing theorems we want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist our objective here is to give stronger results that can be proved by methods that are in some ways more elementary our results are valid in both equal characteristic 0 and in positive prime characteristic p but depend on reduction to characteristic p we use tight closure methods and in consequence we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic for the most basic form of the result all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring every ideal is tightly closed we note that the main argument here is closely related to a proof given in hu 5 14 16 p 45 that regular local rings in characteristic p are ufds

combinatorial symbolic powers abstract symbolic powers are studied in the combinatorial context of monomial ideals when the ideals are generated by quadratic squarefree monomials the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups as a result perfect graphs play an important role in the theory dual to the role played by perfect graphs in the theory of secants of monomial ideals we use grobner degenerations as a tool to reduce questions about symbolic powers of arbitrary ideals to the monomial case among the applications are a new unified approach to the grobner bases of symbolic powers of determinantal and pfaffian ideals

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

identification of miso nonlinear regressions in the presence of a wide class of disturbances a nonparametric algorithm based on multiple fourier series for identification of miso multidimensional input single output systems is investigated it is shown that the algorithm converges for a wide class of noise

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

uniform bounds and symbolic powers on smooth varieties the purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety in particular weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety let

a study of the eigenvalue sensitivity by homotopy and perturbation methods in this work the sensitivity to material characteristics of eigenvalues is studied from an initial structure some defects of material or and geometry are introduced a method is proposed to solve the new eigenvalue problem from the initial one without using classical techniques this method is based on the association of a homotopy transformation and the perturbation method

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

a generic approach for the solution of nonlinear residual equations part ii homotopy and complex nonlinear eigenvalue method abstract solutions of nonlinear smooth pde problems may be approximated as higher order truncated taylor series using the asymptotic numerical method anm the diamant approach already presented in part i is a generic and efficient automatic differentiation implementation of the anm in this second part a diamant based anm driver is designed for the solution of nonlinear problems involving a homotopy that is an artificial transformation of the original problem into a simpler one whose solutions are known complex nonlinear eigenvalue problems are considered as an application the homotopy being achieved by continuation from the real eigenvalue problem to the complex one numerical examples are presented for sandwich beams with frequency dependent viscoelastic cores pvb and 3m isd112 three different constitutive laws a constant modulus a power law and a generalized maxwell model are presented to enhance the capabilities of this diamant driver

symbolic powers of edge ideals abstract let i r k x k x 1 x n be an ideal in a polynomial ring over the field k we define the essential symbolic module of i to be the r i module f i r 2 i r r i where r i i 1 r 1 i i i r i and i m stands for the mth symbolic power of i we will mainly focus on the case where i is generated by square free monomials of degree two among our main results are optimal bounds for the degrees of the minimal generators of f i several criteria for a monomial to be such a generator and an upper bound for the generation type of the symbolic rees algebra of i as a byproduct we recapture the result of simis vasconcelos and villarreal on when such an ideal is normally torsionfree

corrigendum to cohen macaulayness of monomial ideals and symbolic powers of stanley reisner ideals adv math 226 2 2011 1285 1306

regularity of symbolic powers of edge ideals of unicyclic graphs abstract let g be a unicyclic graph with edge ideal i g for any integer s 1 we denote the s th symbolic power of i g by i g s it is shown that reg i g s reg i g s for every s 1