measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

the cohomological dimension of a resultant variety in prime characteristicp 2 2 let f s t s x s q 2 x st q x t 1 f i f 3 be a collection of bii i0 i1 i2 nary quadratic forms with indeterminate coefficients over a field k and let w x r s k x the aim of this paper is to compute the cohomoi j 1f if 3 0 f jf 2 logical dimension of the ideal i r generated by the following sylvester resultants res f f res f f res f f and res f q f f 1 2 1 3 2 3 1 2 3 the closed points of the affine variety defined by i correspond to coefficients of the three forms having a common zero thus in the w x terminology of l1 i is a resultant system for f f and f compare 1 2 3 w x with theorem 1 b in l1 in general if f f are binary forms of 1 s degrees d g d g g d one can show that a resultant system is 1 2 s s w x generated by at least 1 q 1 q d elements theorem 1 in l1 and ts3 t one would like to know if this lower bound is sharp in our case where s s 3 and d s d s d s 2 the question is whether 1 2 3 4 i can be generated by three elements this would imply that h r s 0 i and this is the motivation behind the main result of this paper 4 it turns out that it is possible to show that h r s 0 working in the i subring of joint invariants of g s gl 2 k acting on r via change of variables in s t this calculation is based on some recent results by lyubeznik on f modules which include an algorithm for deciding whether local cohomology modules over regular rings of prime characteristic p vanish to the best of the author s knowledge theorem 9 in this paper is the first application of this algorithm yan has recently produced a 4 w x topological argument showing that h r vanishes in characteristic 0 y i

generators and defining relations for the ring of differential operators on a smooth affine algebraic variety in prime characteristic abstract for the ring of differential operators d o x on a smooth affine algebraic variety x over a perfect field of characteristic p 0 explicit algebra generators and defining relations are found a finite set of generators and a finite set of defining relations are given explicitly for the module der k o x of derivations on the algebra o x of regular functions on the variety x

finiteness properties of local cohomology modules an application of d modules to commutative algebra the main goal of this paper is to establish finiteness properties of local cohomology modules in characteristic 0 that would be analogous to those proven by c huneke and r sharp in characteristicp 0 our method based on the theory of algebraicd modules seems to be the first application ofd modules to commutative algebra

on invariants and endomorphism rings of certain local cohomology modules abstract let r m denote an n dimensional gorenstein ring for an ideal i r with grade i c we define new numerical invariants i j i as the socle dimensions of h m i h i n j r in case of a regular local ring containing a field these numbers coincide with the lyubeznik numbers i j r i we use d d i d dim r i to characterize the surjectivity of the natural homomorphism f r hom r h i r c r h i r c r as a technical tool we study several natural homomorphisms moreover we prove a few results on i j i

local cohomology modules of a smooth z algebra have finitely many associated primes let r be a commutative noetherian ring that is a smooth mathbb z algebra for each ideal mathfrak a of r and integer k we prove that the local cohomology module h k mathfrak a r has finitely many associated prime ideals this settles a crucial outstanding case of a conjecture of lyubeznik asserting this finiteness for local cohomology modules of all regular rings

on the vanishing of local cohomology of the absolute integral closure in positive characteristic abstract the aim of this paper is to extend the main result of c huneke and g lyubeznik in adv math 210 2007 498 504 to the class of rings that are images of cohen macaulay local rings namely let r be a local noetherian domain of positive characteristic that is an image of a cohen macaulay local ring we prove that all local cohomology of r below the dimension maps to zero in a finite extension of the ring as a direct consequence we obtain that the absolute integral closure of r is a big cohen macaulay algebra since every excellent local ring is an image of a cohen macaulay local ring this result is a generalization of the main result of m hochster and huneke in ann of math 135 1992 45 79 with a simpler proof

cofiniteness of local cohomology modules for ideals of small dimension abstract let r be a commutative noetherian ring and let m be a non zero finitely generated r module let i be an ideal of r and t a non negative integer such that dim supp h i i m 1 for all i t it is shown that the r modules h i 0 m h i 1 m h i t 1 m are i cofinite and the r module hom r r i h i t m is finitely generated this immediately implies that if i has dimension one i e dim r i 1 then h i i m is i cofinite for all i 0 this is a generalization of the main results of delfino and marley d delfino t marley cofinite modules and local cohomology j pure appl algebra 121 1997 45 52 and yoshida k i yoshida cofiniteness of local cohomology modules for ideals of dimension one nagoya math j 147 1997 179 191 for an arbitrary noetherian ring r also we prove that if r is local and dim supp h i i m 2 for all i t then the r modules ext r j r i h i i m and hom r r i h i t m are weakly laskerian for all i t and all j 0 as a consequence it follows that the set of associated primes of h i i m is finite for all i 0 whenever dim r i 2

associated primes of local cohomology of flat extensions with regular fibers and σ finite d modules abstract in this manuscript we study the following question raised by mel hochster let r m k be a local ring and s be a flat extension with regular closed fiber is v m s ass s h i i s finite for every ideal i s and i n we prove that the answer is positive when s is either a polynomial or a power series ring over r and dim r i r 1 in addition we analyze when this question can be reduced to the case where s is a power series ring over r an important tool for our proof is the use of finite d modules which are not necessarily finitely generated as d modules but whose associated primes are finite we give examples of this class of d modules and applications to local cohomology

some results on local cohomology of polynomial and formal power series rings the one dimensional case abstract in this paper we prove several results on the finiteness of local cohomology of polynomial and formal power series rings in particular we give a partial affirmative answer for a question of l nunez betancourt 2014 14

lyubeznik numbers of monomial ideals let r k x1 xn be the polynomial ring in n independent variables where k is a field in this work we will study bass numbers of local cohomology modules h r i r supported on a squarefree monomial ideal i r among them we are mainly interested in lyubeznik numbers we build a dictionary between the modules h r i r and the minimal free resolution of the alexander dual ideal i that allow us to interpret lyubeznik numbers as the obstruction to the acyclicity of the linear strands of i the methods we develop also help us to give a bound for the injective dimension of the local cohomology modules in terms of the dimension of the small support

on the support of local cohomology we examine the question of whether the support of an arbitrary local coho mology module of a finitely generated module over a noetherian ring with support in a given ideal must be closed in the zariski topology several results giving an armative answer to this question are given in particular we show that the support is closed when the given ideal has cohomological dimension at most two or the ambient ring is local of dimension at most four

attached primes of local cohomology modules under localization and completion let r m be a noetherian local ring and m a finitely generated r module following i g macdonald 8 the set of all attached primes of the artinian local cohomology module hmi m is denoted by attr hmi m in 13 theorem 3 7 r y sharp proved that if r is a quotient of a gorenstein local ring then the shifted localization principle holds true for any local cohomology modules hmi m i e 1 attrp hprpi dim r p mp qrp q attrhmi m q p for any p spec r in this paper we improve sharp s result as follows the shifted localization principle holds true if and only if r is universally catenary and all its formal fibers are cohen macaulay if and only if the shifted completion principle 2 attr hmi m p attr hmi m assr r pr holds true for any local cohomology module hmi m this also improves the main result of the paper by t d m chau and the first author 2

bass numbers of local cohomology modules let a be a regular local ring of positive characteristic this paper is concerned with the local cohomology modules of a itself but with respect to an arbitrary ideal of a the results include that all the bass numbers of all such local cohomology modules are finite that each such local cohomology module has finite set of associated prime ideals and that whenever such a local cohomology module is artinian then it must be injective this last result had been proved earlier by hartshorne and speiser under the additional assumptions that a is complete and contains its residue field which is perfect the paper ends with some low dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true

cohomological dimension lyubeznik numbers and connectedness in mixed characteristic abstract we establish a second vanishing theorem for local cohomology modules over regular rings of unramified mixed characteristic which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology applying this and new results on the mixed characteristic lyubeznik numbers we further study connectedness properties of the spectra of a certain class of rings

d module structure of local cohomology modules of toric algebras let s be a toric algebra over a field k of characteristic 0 and let i be a monomial ideal of s we show that the local cohomology modules h i i s are of finite length over the ring of differential operators d s k generalizing the classical case of a polynomial algebra s as an application we compute the characteristic cycles of some local cohomology modules

a splitting theorem for local cohomology and its applications abstract let r be a commutative noetherian ring and m a finitely generated r module we show in this paper that for an integer t if the local cohomology module h a i m with respect to an ideal a is finitely generated for all i t then h a i m x m h a i m h a i 1 m for all a filter regular elements x contained in a enough large power of a and all i t 1 as consequences we obtain generalizations by very short proofs of the main results of m brodmann and a l faghani m brodmann a l faghani a finiteness result for associated primes of local cohomology modules proc amer math soc 128 2000 2851 2853 and of h l truong and the first author n t cuong h l truong asymptotic behavior of parameter ideals in generalized cohen macaulay module j algebra 320 2008 158 168

an example of an infinite set of associated primes of a local cohomology module let r m be a local noetherian ring let i subset r be any ideal and let m be a finitely generated r module in 1990 craig huneke conjectured that the local cohomology modules h i i m have finitely many associated primes for all i in this paper i settle this conjecture by constructing a local cohomology module of a local k algebra with an infinite set of associated primes and i do this for any field k

the optimal strategy of a tree classifier abstract this paper deals with the decision rules of a tree classifier for performing the classification at each nonterminal node under the assumption of complete probabilistic information for given tree structure and feature subsets to be used the optimal decision rules strategy are derived which minimize the overall probability of misclassification the primary result is illustrated by an example

associated primes of local cohomology modules and matlis duality abstract let r m be a commutative noetherian local ring of dimension d and i an ideal of r we show that the set of associated primes of the local cohomology module h i 2 r is finite whenever r is regular also it is shown that if x 1 x d is a system of parameters for r then d h x 1 x i i r has infinitely many associated prime ideals for all i d 1 where d hom r e denotes the matlis dual functor and e e r r m is the injective hull of the residue field r m finally we explore a counterexample of grothendieck s conjecture by showing that if d 3 then the r module hom r r i h i d 1 r is not finitely generated where i x 1 x 2 x d

invariant measurement of strange sets in terms of cycles we argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible but is also a theoretically optimal measurement of the invariant properties of a dynamical system

multiple binary decision tree classifiers abstract binary decision trees based on nonparametric statistical models of the data provide a solution to difficult decision problems where there are many classes and many available features related in a complex manner unfortunately the technique requires a very large training set and is often limited by the size of the training set rather than by the discriminatory power of the features this paper demonstrates that higher classification accuracies can be obtained from the same training set by using a combination of decision trees and by reaching a consensus using dempster and shafer s theory of evidence

automatic pattern recognition a study of the probability of error a test sequence is used to select the best rule from a class of discrimination rules defined in terms of the training sequence the vapnik chervonenkis and related inequalities are used to obtain distribution free bounds on the difference between the probability of error of the selected rule and the probability of error of the best rule in the given class the bounds are used to prove the consistency and asymptotic optimality for several popular classes including linear discriminators nearest neighbor rules kernel based rules histogram rules binary tree classifiers and fourier series classifiers in particular the method can be used to choose the smoothing parameter in kernel based rules to choose k in the k nearest neighbor rule and to choose between parametric and nonparametric rules

f injectivity and frobenius closure of ideals in noetherian rings of characteristic p 0 abstract the main aim of this article is to study the relation between f injective singularity and the frobenius closure of parameter ideals in noetherian rings of positive characteristic the paper consists of the following themes including many other topics 1 we prove that if every parameter ideal of a noetherian local ring of prime characteristic p 0 is frobenius closed then it is f injective 2 we prove a necessary and sufficient condition for the injectivity of the frobenius action on h m i r for all i f m r where f m r is the finiteness dimension of r as applications we prove the following results a if the ring is f injective then every ideal generated by a filter regular sequence whose length is equal to the finiteness dimension of the ring is frobenius closed it is a generalization of a recent result of ma and which is stated for generalized cohen macaulay local rings b let r m k be a generalized cohen macaulay ring of characteristic p 0 if the frobenius action is injective on the local cohomology h m i r for all i dim r then r is buchsbaum this gives an answer to a question of takagi 3 we consider the problem when the union of two f injective closed subschemes of a noetherian f p scheme is f injective using this idea we construct an f injective local ring r such that r has a parameter ideal that is not frobenius closed this result adds a new member to the family of f singularities 4 we give the first ideal theoretic characterization of f injectivity in terms the frobenius closure and the limit closure we also give an answer to the question about when the frobenius action on the top local cohomology is injective

finite correlation dimension for stochastic systems with power law spectra abstract we discuss a counter example to the traditional view that stochastic processes lead to a non convergence of the correlation dimension in computed or measured time series specifically we show that a simple class of colored random noises characterized by a power law power spectrum have a finite and predictable value for the correlation dimension these results have implications on the experimental study of deterministic chaos as they indicate that the soie observation of a finite fractal dimension from the analysis of a time series is not sufficient to infer the presence of a strange attractor in the system dynamics we demonstrate that the types of random noises considered herein may be given an interpretation in terms of their fractal properties the consequent exploitation of the non gaussian behavior of these random noises leads us to the introduction of a new time series analysis method which we call multivariate scaling analysis we apply this approach to characterize several global properties of random noise

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

toward an efficient algorithm for deciding the vanishing of local cohomology modules in prime characteristic let r k x 1 dots x n be a ring of polynomials over a field k of characteristic p 0 there is an algorithm due to lyubeznik for deciding the vanishing of local cohomology modules h i i r where i subset r is an ideal this algorithm has not been implemented because its complexity grows very rapidly with the growth of p which makes it impractical in this paper we produce a modification of this algorithm that consumes a modest amount of memory

on the structure of the endomorphism ring of a certain local cohomology module let r m denote an n dimensional gorenstein ring for an ideal i r of height c we are interested in the endomorphism ring b homr hic r hic r it turns out that b is a commutative ring in the case of r m a regular local ring containing a field b is a cohen macaulay ring its properties are related to the highest lyubeznik number l dimkextrd k hic r d dimr i in particular r b if and only if l 1 moreover we show that the natural homomorphism extrd k hic r k is non zero

propagation of periodic and chaotic action potential trains along nerve fibers abstract in this paper we report the findings that the action potential trains transmitted along the nerve fibers are encoded not only periodically but also chaotically first spontaneous action potentials along a single fiber of injured sciatic nerves in the anesthetized rat were recorded then the data were divided into two groups and analyzed with different methods phase space representation spectral analysis and the calculation of correlation dimension were used for the first group of data sampled with constant frequency due to the serious influence of the measurement noise no reliable conclusion can be drawn from them for the second group of data of the interspike intervals isi which seem to convey more rich and important information nonlinear forecasting method the surrogate data and the plot of isi n 1 vs isi n were used in the analysis good results have been obtained which confirm with those from the cell model the largest lyapunov exponent lle were calculated not only to further support our findings of chaos but also to quantitatively determine the degree of chaos

periodic orbits as the skeleton classical and quantum chaos a description of a low dimensional deterministic chaotic system in terms of unstable periodic orbits cycles is a powerful tool for theoretical and experimental analysis of both classical and quantum deterministic chaos comparable to the familiar perturbation expansions for nearly integrable systems the infinity of orbits characteristic of a chaotic dynamical system can be resummed and brought to a selberg product form dominated by the short cycles and the eigenvalue spectrum of operators associated with the dynamical flow can then be evaluated in terms of unstable periodic orbits methods for implementing this computation for finite subshift dynamics are introduced

the d module structure of r f modules let r be a regular ring essentially of finite type over a perfect field k an r module m is called a unit r f module if it comes equipped with an isomorphism f e m m where f denotes the frobenius map on spec r and f e is the associated pullback functor it is well known that m then carries a natural d r module structure in this paper we investigate the relation between the unit r f structure and the induced d r structure on m in particular it is shown that if k is algebraically closed and m is a simple finitely generated unit r f module then it is also simple as a d r module an example showing the necessity of k being algebraically closed is also given

ideals in a perfect closure linear growth of primary decompositions and tight closure this paper is concerned with tight closure in a commutative noetherian ring r of prime characteristic p and is motivated by an argument of k e smith and i swanson that shows that if the sequence of frobenius powers of a proper ideal a of r has linear growth of primary decompositions then tight closure of a commutes with localization at the powers of a single element it is shown in this paper that provided r has a weak test element linear growth of primary decompositions for other sequences of ideals of r that approximate in a certain sense the sequence of frobenius powers of a would not only be just as good in this context but in the presence of a certain additional finiteness property would actually imply that tight closure of a commutes with localization at an arbitrary multiplicatively closed subset of r work of m katzman on the localization problem for tight closure raised the question as to whether the union of the associated primes of the tight closures of the frobenius powers of a has only finitely many maximal members this paper develops through a careful analysis of the ideal theory of the perfect closure of r strategies for showing that tight closure of a specified ideal a of r commutes with localization at an arbitrary multiplicatively closed subset of r and for showing that the union of the associated primes of the tight closures of the frobenius powers of a is actually a finite set several applications of the strategies are presented in most of them it was already known that tight closure commutes with localization but the resulting affirmative answers to katzman s question in the various situations considered are believed to be new

tree classifier design with a permutation statistic abstract this paper proposes a permutation statistic as a basis for designing binary tree classifiers of d dimensional patterns the statistic permits an objective and easily interpreted evaluation of a feature threshold pair that is the core of the design paradigm it has a known distribution that is independent of the number of training patterns after defining the permutation statistic and explaining its properties the paper proposes the design procedure whereby each non terminal node is assigned a feature and a threshold on that feature the design procedure is demonstrated on six test sets and compared to designs based on mutual information

cartier modules finiteness results on a locally noetherian scheme x over a field of positive characteristic p we study the category of coherent ox modules m equipped with a pe linear map i e an additive map c ox ox satisfying rc m c r p e m for all m a m r a ox the notion of nilpotence meaning that some power of the map c is zero is used to rigidify this cate gory the resulting quotient category called cartier crystals satisfies some strong finiteness conditions the main result in this paper states that if the frobenius morphism on x is a finite map i e if x is f finite then all cartier crystals have finite length we further show how this and related results can be used to recover and generalize other finiteness results of hartshorne speiser 19 lyubeznik 27 sharp 30 enescu hochster 14 and hochster 20 about the structure of modules with a left action of the frobenius for example we show that over any regular f finite scheme x lyubeznik s f finite modules 27 have finite length

random hyperplane search trees a hyperplane search tree is a binary tree used to store a set s of n d dimensional data points in a random hyperplane search tree for s the root represents a hyperplane defined by d data points drawn uniformly at random from s the remaining data points are split by the hyperplane and the definition is used recursively on each subset we assume that the data are points in general position in mathbb r d we show that uniformly over all such data sets s the expected height of the hyperplane tree is not worse than that of the k d tree or the ordinary one dimensional random binary search tree and that for any fixed d ge3 the expected height improves over that of the standard random binary search tree by an asymptotic factor strictly greater than one

asymptotic behavior of parameter ideals in generalized cohen macaulay modules abstract the purpose of this paper is to give affirmative answers to two open questions as follows let r m be a generalized cohen macaulay noetherian local ring both questions the first question was raised by m rogers m rogers the index of reducibility for parameter ideals in low dimension j algebra 278 2004 571 584 and the second one is due to s goto and h sakurai s goto h sakurai the equality i 2 q i in buchsbaum rings rend sem mat univ padova 110 2003 25 56 ask whether for every parameter ideal q contained in a high enough power of the maximal ideal m the following statements are true 1 the index of reducibility n r q r is independent of the choice of q and 2 i 2 q i where i q r m

a cohomological stability result for projective schemes over surfaces let x x0 be a projective morphism of schemes such that x0 is noetherian and essentially of finite type over a field k let i n0 let f be a coherent sheaf of modules and let l be an ample invertible sheaf over x we show that the set of associated points of the higher direct image sheaf ultimately becomes constant if n tends to provided x0 has dimensione 2 if this stability result need not hold any more to prove this we show that the set of associated primes of the n th graded component of the i th local cohomology module of a finitely generated graded module m over a homogeneous noetherian ring which is essentially of finite type over a field becomes ultimately constant in codimension 2 if n tends to

bockstein homomorphisms in local cohomology let r be a polynomial ring in finitely many variables over the integers fix an ideal a of r we prove that for all but finitely prime integers p the bockstein homomorphisms on local cohomology h a r pr h k 1 a r pr are zero this vanishing of bockstein homomorphisms is predicted by lyubeznik s conjecture which states that when r is a regular ring the modules h a r have finitely many associated prime ideals we further show that when r is replaced by a hypersurface ring the vanishing of bockstein homomorphisms as well as the analogue of lyubeznik s conjecture need not hold bockstein homomorphisms have their origins in algebraic topology we establish a connection between algebraic and topological bockstein homomorphisms through stanley reisner theory

an iterative growing and pruning algorithm for classification tree design a critical issue in classification tree design obtaining right sized trees i e trees which neither underfit nor overfit the data is addressed instead of stopping rules to halt partitioning the approach of growing a large tree with pure terminal nodes and selectively pruning it back is used a new efficient iterative method is proposed to grow and prune classification trees this method divides the data sample into two subsets and iteratively grows a tree with one subset and prunes it with the other subset successively interchanging the roles of the two subsets the convergence and other properties of the algorithm are established theoretical and practical considerations suggest that the iterative free growing and pruning algorithm should perform better and require less computation than other widely used tree growing and pruning algorithms numerical results on a waveform recognition problem are presented to support this view

an excellent f pure ring of prime characteristic has a big tight closure test element in two recent papers the author has developed a theory of graded annihilators of left modules over the frobenius skew polynomial ring over a commutative noetherian ring r of prime characteristic p and has shown that this theory is relevant to the theory of test elements in tight closure theory one result of that work was that if r is local and the r module structure on the injective envelope e of the simple r module can be extended to a structure as a torsion free left module over the frobenius skew polynomial ring then r is f pure and has a tight closure test element one of the central results of this paper is the converse namely that if r is f pure then e has a structure as a torsion free left module over the frobenius skew polynomial ring a corollary is that every f pure local ring of prime characteristic even if it is not excellent has a tight closure test element these results are then used along with embedding theorems for modules over the frobenius skew polynomial ring to show that every excellent not necessarily local f pure ring of characteristic p must have a so called big test element

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

d module and f module length of local cohomology modules let r be a polynomial or power series ring over a field k we study the length of local cohomology modules hji r in the category of d modules and fmodules we show that the d module length of hji r is bounded by a polynomial in the degree of the generators of i in characteristic p 0 we obtain upper and lower bounds on the f module length in terms of the dimensions of frobenius stable parts and the number of special primes of local cohomology modules of r i the obtained upper bound is sharp if r i is an isolated singularity and the lower bound is sharp when r i is gorenstein and f pure we also give an example of a local cohomology module that has different d module and f module lengths

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

on the set of associated primes of a local cohomology module abstract assume r is a local cohen macaulay ring it is shown that ass r h l i r is finite for any ideal i and any integer l provided ass r h 2 x y r is finite for any x y r and ass r h 3 x 1 x 2 y r is finite for any y r and any regular sequence x 1 x 2 r furthermore it is shown that ass r h l i r is always finite if dim r 3 the same statement is even true for dim r 4 if r is almost factorial

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

constructing arbitrary torsion elements for a local cohomology module local cohomology modules are in general not finitely generated over the ambient ring towards a better understanding of the structure of these modules huneke 1992 hu asked if they have finitely many associated prime ideals the answer is negative several authors have constructed a ring whose local cohomology module has infinitely many associated primes in certain situations one can prove that a local cohomology module has infinitely many associated primes by showing that z pz embeds into the local cohomology module for each prime integer p we show that in fact each finitely generated abelian group embeds into a local cohomology module