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

minimal resolutions of algebras abstract a method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself the method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal of relations that is either homogeneous or admissible with some additional finiteness restrictions in the latter case in particular it applies to any finite dimensional algebra over an algebraically closed field the method is illustrated by a number of examples viz truncated algebras monomial algebras and koszul algebras with the aim of unifying existing treatments of these in the literature

m koszul artin schelter regular algebras abstract this paper studies the homological determinants and nakayama automorphisms of not necessarily noetherian m koszul twisted calabi yau or equivalently m koszul artin schelter regular algebras dubois violette showed that such an algebra is isomorphic to a derivation quotient algebra d w i for a unique up to scalar multiples twisted superpotential w by definition d w i is the quotient of the tensor algebra tv where v d w i 1 by i w the ideal generated by all i th order left partial derivatives of w the restriction map v is used to identify the group of graded algebra automorphisms of d w i with a subgroup of gl v we show that the homological determinant of a graded algebra automorphism of an m koszul artin schelter regular algebra d w i is given by the formula hdet w m i w it follows from this that the homological determinant of the nakayama automorphism of an m koszul artin schelter regular algebra is 1 as an application we prove that the homological determinant and the usual determinant coincide for most quadratic noetherian artin schelter regular algebras of dimension 3

nonlinear analysis of river flow time sequences within the field of chaos theory several methods for the analysis of complex dynamical systems have recently been proposed in light of these ideas we study the dynamics which control the behavior over time of river flow investigating the existence of a low dimension deterministic component the present article follows the research undertaken in the work of porporato and ridolfi 1996a in which some clues as to the existence of chaos were collected particular emphasis is given here to the problem of noise and to nonlinear prediction with regard to the latter the benefits obtainable by means of the interpolation of the available time series are reported and the remarkable predictive results attained with this nonlinear method are shown

a algebra structures associated to k2 algebras the notion of a k2 algebra was recently introduced by cassidy and shelton as a generalization of the notion of a koszul algebra the yoneda algebra of any connected graded algebra admits a canonical a algebra structure this structure is trivial if the algebra is koszul we study the a structure on the yoneda algebra of a k2 algebra for each non negative integer n we prove the existence of a k2 algebra b and a canonical a algebra structure on the yoneda algebra of b such that the higher multiplications mi are nonzero for all 3 i n 3 we also provide examples which show that the k2 property is not detected by any obvious vanishing patterns among higher multiplications

a deterministic geometric representation of temporal rainfall results for a storm in boston the use of a deterministic fractal multifractal fm representation to model high resolution rainfall time series via projections of fractal interpolating functions weighed by multifractal measures is reported it is shown that the intrinsic shape and variability of an 8 hour boston storm recorded every 15 s on october 25 1980 may be encoded wholistically employing the fractal geometric methodology it is illustrated that the fm methodology provides very faithful descriptions of both major trends and small noisy fluctuations for this storm resulting in preservation of not only classical statistical characteristics of the records but also multifractal and chaotic properties present in them these results and those for other storms suggest that a stochastic framework for rainfall may be bypassed in favor of a deterministic representation based on projections

higher symplectic reflection algebras and non homogeneous n koszul property abstract we study a class of associative algebras associated to finite groups acting on a vector space these algebras are non homogeneous n koszul algebra generalizations of symplectic reflection algebras p etingof v ginzburg symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism invent math 147 2002 243 348 the koszul property was generalized to homogeneous algebras of degree n 2 in r berger koszulity for nonquadratic algebras j algebra 239 2001 705 734 in the present paper the extension of the koszul property to non homogeneous algebras is realized through a pbw theorem this pbw theorem is the generalization to the n case of a quadratic result obtained by braverman and gaitsgory a braverman d gaitsgory poincare birkhoff witt theorem for quadratic algebras of koszul type j algebra 181 1996 315 328 see also a polishchuk l positselski quadratic algebras univ lecture ser vol 37 amer math soc providence ri 2005 we work in the general setting where the ground ring is an arbitrary von neumann regular ring

generalized koszul properties for augmented algebras abstract under certain conditions a filtration on an augmented algebra a admits a related filtration on the yoneda algebra e a ext a k k we show that there exists a bigraded algebra monomorphism gr e a e gr gr a where e gr gr a is the graded yoneda algebra of gra this monomorphism can be applied in the case where a is connected graded to determine that a has the k 2 property recently introduced by cassidy and shelton

superpotentials calabi yau algebras and pbw deformations abstract the paper 9 by bocklandt schedler and wemyss considers path algebras with relations given by the higher derivations of a superpotential giving a condition for such an algebra to be calabi yau in particular they show that the algebra c v g for v a finite dimensional c vector space and g a finite subgroup of gl v is morita equivalent to a path algebra with relations given by a superpotential and is calabi yau for g sl v in this paper we extend these results giving a condition for a pbw deformation of a calabi yau koszul path algebra with relations given by a superpotential to have relations given by a superpotential and proving these are calabi yau in certain cases we apply our methods to symplectic reflection algebras where we show that every symplectic reflection algebra is morita equivalent to a path algebra whose relations are given by the higher derivations of an inhomogeneous superpotential in particular we show these are calabi yau regardless of the deformation parameter also for g a finite subgroup of gl 2 c not contained in sl 2 c we consider pbw deformations of a path algebra with relations which is morita equivalent to c x y g we show there are no non trivial pbw deformations when g is a small subgroup

higher koszul algebras and a infinity algebras abstract we study a class of a algebras named 2 p algebras which is related to the class of p homogeneous algebras especially to the class of p koszul algebras a general method to construct 2 p algebras is given koszul dual of a connected graded algebra is defined in terms of a algebra it is proved that a p homogeneous algebra a is p koszul if and only if the koszul dual e a is a reduced 2 p algebra and generated by e 1 a the 2 p algebra structure of the koszul dual e a of a p koszul algebra a is described explicitly a necessary and sufficient condition for a p homogeneous algebra to be a p koszul algebra is also given when the higher multiplications on the koszul dual are ignored

finite generation of ext for a generalization of d koszul algebras in this paper we study the finite generation of ext algebras of a class of algebras called resolution determined algebras we characterize the resolution determined algebras which are monomial algebras if is a graded algebra such that the associated monomial algebra is resolution determined we classify when the ext algebra of is finitely generated

killing of supports on graded algebras killing of supports along subsets u of a group g and regradings along certain maps of groups g g are studied in the context of group graded algebras we show that under precise conditions on u and the module theories over the initial and the final algebras are functorially well connected special attention is paid to g z in which case the results can be applied to n koszul algebras

multilinear forms and graded algebras abstract in this paper we investigate the class of the connected graded algebras which are finitely generated in degree 1 which are finitely presented with relations of degrees greater or equal to 2 and which are of finite global dimension d and gorenstein for d greater or equal to 4 we add the condition that these algebras are homogeneous and koszul it is shown that each such algebra is completely characterized by a multilinear form satisfying a twisted cyclicity condition and some other nondegeneracy conditions depending on the global dimension d this multilinear form plays the role of a volume form and canonically identifies in the quadratic case to a nontrivial hochschild cycle of maximal degree several examples including the yang mills algebra and the extended 4 dimensional sklyanin algebra are analyzed in this context actions of quantum groups are also investigated

hopf actions and nakayama automorphisms abstract let h be a hopf algebra with antipode s and let a be an n koszul artin schelter regular algebra we study connections between the nakayama automorphism of a and s 2 of h when h coacts on a inner faithfully several applications pertaining to hopf actions on artin schelter regular algebras are given

third derivative modification of k step block falkner methods for the numerical solution of second order initial value problems abstract this paper is devoted to the development and analysis of a modified family of falkner type methods for solving differential systems of second order initial value problems the approaches of collocation and interpolation are adopted to derive the new methods these modified methods are implemented in block form to obtain the numerical solutions to the considered problems the study of the properties of the proposed block falkner type methods reveals that they are consistent and zero stable and thus convergent from the stability analysis it could be seen that the proposed falkner methods have non empty stability regions for k 2 3 4 some numerical test are presented to illustrate the efficiency of the proposed family

chaos in rainfall the question is posed regarding possible chaotic dynamics in the temporal rainfall of storm events chaotic dynamics implies a nonlinear deterministic system very sensitive to initial conditions which yields outputs indistinguishable from a stochastic process by standard techniques the trajectories of these systems in the phase space are characterized by being contained in a strange attractor of fractal dimension a time series of 1990 points of 15 s rainfall for the storm of october 25 1980 in boston is analyzed in detail toward this objective it is found that both the characteristics of the correlation integral and the lyapunov exponents of the historical data give preliminary support to the presence of chaotic dynamics with a strange attractor the implications of these findings for the modeling of storm rainfall and for the limits of predictability of the process are discussed jointly with the research challenges lying ahead in this field

confluence and koszulity abstract a lattice theoretic approach to koszulity is well known this paper gives a lattice theoretic approach to polynomiality i e existence of a p b w basis in priddy s sense this provides a straightforward proof for priddy s theorem polynomiality is interpreted in terms of confluence in noncommutative computational algebra the basic notion of a reduction operator is investigated in some detail the reduction algebras are introduced to study the confluence of two reduction operators in terms of representation theory the meet and the join of two reduction operators are constructed in an algorithmic way a geometric characterization of the confluence is obtained

convergent presentations and polygraphic resolutions of associative algebras several constructive homological methods based on noncommutative grobner bases are known to compute free resolutions of associative algebras in particular these methods relate the koszul property for an associative algebra to the existence of a quadratic grobner basis of its ideal of relations in this article using a higher dimensional rewriting theory approach we give several improvements of these methods we define polygraphs for associative algebras as higher dimensional linear rewriting systems that generalise the notion of noncommutative grobner bases and allow more possibilities of termination orders than those associated to monomial orders we introduce polygraphic resolutions of associative algebras giving a categorical description of higher dimensional syzygies for presentations of algebras we show how to compute polygraphic resolutions starting from a convergent presentation and how these resolutions can be linked with the koszul property

koszul calculus for n homogeneous algebras abstract we extend koszul calculus defined on quadratic algebras by berger et al 2018 9 to n homogeneous algebras for any n 2 quadratic algebras corresponding to n 2 we emphasize that n homogeneous algebras are considered in full generality with no koszulity assumption koszul cup and cap products are introduced and are reduced to usual cup and cap products if n 2 but if n 2 they are defined by very specific expressions these specific expressions are compatible with the koszul differentials and provide associative products on classes there is no associativity in general on chains cochains suggesting that koszul cochains should constitute an a algebra acting as an a bimodule on koszul chains

scaling characteristics in the taiwan stock market some statistical tools including histogram spectral analysis and fractal theory were used on the daily taiwan stock price index tspi from 1987 to 2002 to examine the possible scale invariant behavior and the clustering characteristics in taiwan stock market it was found that the tspi data exhibited the characteristic of right skewed frequency distribution the long term memory and the possibility of scale invariance were roughly identified through the analysis of autocorrelation and power spectrum respectively the monofractal analysis was then performed by the box counting method scale invariance was clearly found in the time series and the box dimension was shown to be a decreasing function of the threshold index level implying multifractal characteristics i e the low and high regions scale differently to test this hypothesis the time series were transferred into a useful compact form through the multifractal formalism namely the q q and f plots the analysis confirmed the existence of multifractal characteristics in the investigated time series the origin of multifractal phenomena in taiwan stock market might be interpreted in terms of the multiplicative cascade process of stock market information

support vector machines for nonlinear state space reconstruction application to the great salt lake time series 1 the reconstruction of low order nonlinear dynamics from the time series of a state variable has been an active area of research in the last decade the 154 year long biweekly time series of the great salt lake volume has been analyzed by many researchers from this perspective in this study we present the application of a powerful state space reconstruction methodology using the method of support vector machines svm to this data set svm are machine learning systems that use a hypothesis space of linear functions in a kernel induced higher dimensional feature space svm are optimized by minimizing a bound on a generalized error risk measure rather than just the mean square error over a training set under mercer s conditions on the kernels the corresponding optimization problems are convex hence global optimal solutions can be readily computed the svm based reconstruction is used to develop time series forecasts for multiple lead times ranging from 2 weeks to several months unlike previously reported methodologies svm are able to extract the dynamics using only a few past observed data points out of the training examples the reliability of the algorithm in learning and forecasting the dynamics is tested using split sample sensitivity analysis with a particular interest in forecasting extreme states efforts are also made to assess variations in predictability as a function of initial conditions and as a function of the degree of extrapolation from the state space used for learning the model

pbw deformations of n koszul algebras abstract for a quotient algebra u of the tensor algebra we give explicit conditions on its relations for u being a pbw deformation of an n koszul algebra a we show that there is a one to one correspondence between such deformations and a class of a structures on the yoneda algebra ext a k k of a we compute the pbw deformations of the algebra whose relations are the anti symmetrizers of degree n and also of cubic artin schelter algebras

a p stable complete in phase obrechkoff trigonometric fitted method for periodic initial value problems in this paper we derive a p stable trigonometric fitted obrechkoff method with phase lag frequency distortion infinity it is easy to see from numerical results presented that the new method is much more accurate than previous methods

homologies of cubic artin schelter regular algebras abstract the hochschild homology of cubic artin schelter regular algebras of type a with generic coefficients is computed we follow the method used by van den bergh k theory 8 1994 213 230 in the quadratic case and consider these algebras as deformations of a polynomial algebra with remarkable poisson brackets a new morphism of resolutions is introduced the de rham cohomology cyclic and periodic cyclic homologies and the hochschild cohomology are also computed

a duality theorem for generalized koszul algebras abstract we show that if is a n koszul algebra and e e is its yoneda algebra then there is a full subcategory l e of the category gr e of graded e modules which contains all the graded e modules presented in even degrees that embeds fully faithfully in the category c gr of cochain complexes of graded modules that extends the known equivalence for koszul i e n 2 between gr e and the category of linear complexes of graded modules

an attractor in a solar time series a time series analysis of observed solar radio pulsations suggests that there must be a low dimensional attractor the power spectrum cannot be interpreted as a superposition of periodic components estimates of the maximum lyapunov exponent and of the kolmogorov entropy give some indications for a deterministic chaos in order to study the limitations inherent in small data samples we include data from the lorenz model and an artificial noise record consequences for the physical modelling of the pulsation event are discussed

fractal nature of highway traffic data in this paper we applied a fractal approach to analyze the traffic data collected from the beijing yuquanying the power spectrum the empirical probability distribution function the statistical moment scaling function and the autocorrelation function are used as indicators to investigate the presence of the fractal the results from the fractal identification methods indicate that these data exhibit fractal behavior a fractal framework seemed well suited for description of the data observed here but its suitability for general traffic systems was not clear

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

p stable symmetric super implicitmethods for periodic initial value problems the idea of super implicit methods requiring not just past and present but also future values was suggested by fukushima recently here we construct p stable super implicit methods for the solution of second order initial value problems the benefit of such methods is realized when using vector or parallel computers

pbw deformation theory and regular central extensions a deformation u of a graded k algebra a is said to be of pbw type if gru is a it has been shown for koszul and n koszul algebras that the deformation is pbw if and only if the relations of u satisfy a jacobi type condition in particular for these algebras the determination of the pbw property is a finite and explicitly determined linear algebra problem we extend these results to an arbitrary graded k algebra using the notion of central extensions of algebras and a homological constant attached to a which we call the complexity of a

koszulity for nonquadratic algebras abstract it is known that a koszul algebra is defined as being a quadratic algebra with a pure resolution of the ground field in this paper we extend koszulity to algebras whose relations are homogeneous of degree s 2 a cubic artin schelter regular algebra has motivated our work generalized koszulity is connected to lattice distributivity and to confluence a generalized symmetric algebra is proved to be generalized koszul and the bimodule version of the generalized koszul resolution is used for investigating its hochschild homology

p stable high order super implicit and obrechkoff methods for periodic initial value problems this paper discusses the numerical solution of periodic initial value problems two classes of methods are discussed super implicit and obrechkoff the advantage of obrechkoff methods is that they are high order one step methods and thus will not require additional starting values on the other hand they will require higher derivatives of the right hand side in cases when the right hand side is very complex we may prefer super implicit methods we develop a super implicit p stable method of order 12 and obrechkoff method of order 18

on the definition of multi koszul modules abstract in 8 we introduced the notion of multi koszul algebra it is an extension of the definition of generalized koszul algebra given by r berger in 1 for homogeneous algebras see also 7 that can be applied to any nonnegatively graded connected algebra over a field k the goal of this article is on the one hand to properly extend the notion of multi koszul algebra to the case where the base ring k is a product of copies of a field k which a priori allows us to treat quiver algebras and on the other hand to introduce the notion of multi koszul module such that it extends the usual definition of generalized koszul module over a generalized koszul algebra we show eventually that multi koszul algebras and multi koszul modules are strongly linked via the notion of one point extensions as in the case of generalized koszul algebras moreover we describe the complete structure of right a module on e x t a m k over e x t a k k where m is a multi koszul module over a multi koszul algebra a extending a result in 9 as a corollary we obtain that the underlying right module structure of e x t a m k over e x t a k k is generated by the component of cohomological degree zero as in the case of generalized koszul modules over generalized koszul algebras

predictive uncertainty of chaotic daily streamflow using ensemble wavelet networks approach 1 perfect or even mediocre weather predictions over a long period are almost impossible because of the ultimate growth of a small initial error into a significant one even though the sensitivity of initial conditions limits the predictability in chaotic systems an ensemble of prediction from different possible initial conditions and also a prediction algorithm capable of resolving the fine structure of the chaotic attractor can reduce the prediction uncertainty to some extent all of the traditional chaotic prediction methods in hydrology are based on single optimum initial condition local models which can model the sudden divergence of the trajectories with different local functions conceptually global models are ineffective in modeling the highly unstable structure of the chaotic attractor this paper focuses on an ensemble prediction approach by reconstructing the phase space using different combinations of chaotic parameters i e embedding dimension and delay time to quantify the uncertainty in initial conditions the ensemble approach is implemented through a local learning wavelet network model with a global feed forward neural network structure for the phase space prediction of chaotic streamflow series quantification of uncertainties in future predictions are done by creating an ensemble of predictions with wavelet network using a range of plausible embedding dimensions and delay times the ensemble approach is proved to be 50 more efficient than the single prediction for both local approximation and wavelet network approaches the wavelet network approach has proved to be 30 50 more superior to the local approximation approach compared to the traditional local approximation approach with single initial condition the total predictive uncertainty in the streamflow is reduced when modeled with ensemble wavelet networks for different lead times localization property of wavelets utilizing different dilation and translation parameters helps in capturing most of the statistical properties of the observed data the need for taking into account all plausible initial conditions and also bringing together the characteristics of both local and global approaches to model the unstable yet ordered chaotic attractor of a hydrologic series is clearly demonstrated

an improved trigonometrically fitted p stable obrechkoff method for periodic initial value problems in this paper we present an improved p stable trigonometrically fitted obrechkoff method with phase lag frequency distortion infinity compared with the previous p stable trigonometrically fitted obrechkoff method developed by simos our new method is simpler in structure and more stable in computation we have also improved the accuracy of the first order derivative formula from the numerical illustration presented we can show that the new method is much more accurate than the previous methods

representations of non commutative quantum groups kriegk benoit univ st etienne fac sci tech math lab f 42023 st etienne 02 france van den bergh michel univ hasselt b 3590 diepenbeek belgium

representations of yang mills algebras the aim of this article is to describe families of representations of the yang mills algebras ym n n 2 n 2 dened by a connes and m duboisviolette we rst describe some irreducible nite dimensional representations next we provide families of innite dimensional representations of ym big enough to separate points of the algebra in order to prove this result we prove and use that all weyl algebras ar k are epimorphic images of ym n

is correlation dimension a reliable indicator of low dimensional chaos in short hydrological time series the reliability of the correlation dimension estimation in short hydrological time series is investigated using an inverse approach according to this approach first predictions are made using the phase space reconstruction technique and the artificial neural networks the correlation dimension is estimated next independently and is compared with the prediction results a short hydrological series monthly runoff series of 48 years with a total of only 576 values observed at the coaracy nunes araguari river watershed in northern brazil is studied the correlation dimension results are in reasonably good agreement with the optimal embedding dimension obtained from the phase space method and the optimal number of inputs from the neural networks no underestimation of the correlation dimension is observed due to the small data size rather there seems to be a slight overestimation due to the presence of noise in the data the results indicate that the accuracy of the correlation dimension may not be judged on the basis of the length of the time series but on whether the time series is long enough to reasonably represent the dynamical changes in the system such an observation suggests that the correlation dimension could indeed be a reliable indicator of low dimensional chaos even in short hydrological time series which is certainly encouraging news for hydrologists who often have to deal with short time series less

biases of correlation dimension estimates of streamflow data in the canadian prairies the correlation dimension d2 of the unregulated streamflow data of six rivers in the canadian prairies were estimated using the hill estimate and grassberger procaccia algorithms from both procedures and using an embedding phase space up to 10 the d2 obtained for all cases are found to saturate at 3 0 however the saturated d2 generally increased to between 4 and 6 when the data were first randomly resampled prior to estimating the correlation dimension since the highest embedding dimension d used in this study is 10 theoretically randomly resampled data should have a d2 approximately equal to 10 therefore there is a consistent underestimation of d2 by an amount of 4 6 by accounting for the combined clustering effect of the streamflow data s underlying distribution which is gamma the effect of sample size and possibly the choice of time delay the d2 of the prairie streamflow data should be 7 9 the hypothesis that the clustering effect of unevenly distributed data contributed to the underestimation of d2 was confirmed from the different d2 obtained for random numbers of uniform gamma and poisson distributions in addition analysis of the uniformly distributed random numbers and streamflow data showed that increasing the sample size from say 6500 to 17 000 marginally improved the estimated d2 however for the range of time delay chosen for this study 40 180 days its effect on d2 is less obvious this may be partly because all the streamflow data tested exhibit long term persistence

small scale turbulence in the plankton low order deterministic chaos or high order stochasticity abstract almost all regions of the oceans are heavily influenced by the effects of physical processes such as turbulence which in turn influence the distribution and ecology of organisms that occupy these regions there is a real need for additional focus placed on the precise knowledge of both physical and biological processes which is often difficult using basic time series analysis in that way we applied nonlinear analysis techniques to high frequency time series of temperature salinity and phytoplankton concentrations recorded in different hydrodynamical regimes related to tidal forcing in a tidally mixed coastal ecosystem techniques devoted to the identification of low order deterministic chaos cannot find evidence of chaos while a lower dimensionality was identified in low hydrodynamic conditions the results rather suggest stochastic time series with many degrees of freedom no obvious attractor in phase space trajectory lle indistinguishable from zero absence of convergence of the correlation integral we then applied to these data specific multifractal analysis techniques and showed that these time series clearly exhibit high order stochasticity in addition the stochastic structure of purely passive scalars i e temperature and salinity remained invariant while the one of phytoplankton biomass must be regarded as highly structured in time by both hydrodynamic and advective processes

obrechkoff versus super implicit methods for the solution of first and second order initial value problems abstract this paper discusses the numerical solution of first order initial value problems and a special class of second order ones those not containing first derivative two classes of methods are discussed super implicit and obrechkoff we will show equivalence of super implicit and obrechkoff schemes the advantage of obrechkoff methods is that they are high order one step methods and thus will not require additional starting values on the other hand they will require higher derivatives of the right hand side in case the right hand side is complex we may prefer super implicit methods the disadvantage of super implicit methods is that they in general have a larger error constant to get the same error constant we require one or more extra future values we can use these extra values to increase the order of the method instead of decreasing the error constant one numerical example shows that the super implicit methods are more accurate than the obrechkoff schemes of the same order

multiderivative methods for periodic initial value problems a family of two step multiderivative methods is developed for the numerical solution of periodic initial value problems of the form bf y t bf f t bf y with initial conditions bf y t 0 bf y 0 bf y t 0 bf y 0 given the methods are developed by showing that the solution of the single test equation y t lambda 2 y t lambda real satisfies a recurrence relation in which exponential functions are involved the methods of the family are then derived by replacing the exponential terms by pade approximants the methods are analyzed for accuracy and periodicity properties the error constants and intervals of periodicity are contained in an appendix those pade approximants for which the degree of the numerator do not exceed the degree of the denominator are seen to be p stable suitable predictor corrector combinations of the methods are analyzed for use in pece mode the methods are compared with existing methods and they are tested on four problems from the

on the homology of associative algebras we present a new free resolution for k as an g module where g is an associative augmented algebra over a field k the resolution reflects the combinatorial properties of g introduction let k be a field and let g be an associative augmented k algebra for many purposes one wishes to have a projective resolution of k as a g module the bar resolution is always easy to define but it is often too large to use in practice at the other extreme minimal resolutions may exist but they are often hard to write down in a way that is amenable to calculations the main theorem of this paper presents a compromise resolution though rarely minimal it is small enough to offer some bounds but explicit enough to facilitate calculations as it relies heavily upon combinatorial constructions it is best suited for analyzing otherwise tricky algebras given via generators and relations since several results we get as consequences of the main theorem have been obtained before through other means this paper may be viewed as generalizing and unifying several seemingly unrelated ideas in particular we are generalizing priddy s results on koszul algebras 12 extending homology computations by govorov 9 and backelin 3 and complementing bergman s methods regarding the diamond lemma 6 three results may be of interest the homology of the mod p steenrod algebra is given in terms of the homology of a new chain complex smaller than the a algebra in theorem 3 5 formula 16 offers an efficient algorithm for the determination of hilbert series and theorem 4 2 asserts the existence of new bounds on the torsion groups of commutative graded rings 1 definitions and the main theorem throughout this paper k denotes any field and g is an associative k algebra with unity the field k embeds in g via 77 k g and we suppose that g has an augmentation i e a k algebra map s g k for which 77 is a right inverse s denotes a set of generators for g as a k algebra and k s is the free associative k algebra with unity on s there is a canonical surjection f k s g once s is chosen and the augmentation e is determined once we know s x for each x e s in particular this means that k s may be augmented such that f becomes a map of augmented algebras to s we associate a function e s z called a grading in the absence of a more compelling choice we often take e to be grading by length i e e x 1 for received by the editors may 23 1983 and in revised form february 22 1984 this paper was the subject of an invited one hour address in boulder colorado during the week long ams summer program on combinatorics and algebra june 1983 1980 mathematics subject classification 1985 revision primary 16a62 secondary 13d03 55s10 1986 american mathematical society 0002 9947 86 1 00 25 per page

generalizing the notion of koszul algebra we introduce a generalization of the notion of a koszul algebra which includes graded algebras with relations in different degrees and we establish some of the basic properties of these algebras this class is closed under twists twisted tensor products regular central extensions and ore extensions we explore the monomial algebras in this class and we include some well known examples of algebras that fall into this class

scaling characteristics in aftershock sequence of earthquake abstract the possible scale invariant behavior and the clustering characteristics in aftershock sequence of chi chi taiwan main earthquake asccme that occurred in 1999 9 20 17 47 were investigated by means of some statistical tools histogram spectral analysis and fractal theory the examined data were constructed from the aftershocks that occurred at the locations defined at longitude 120 1 121 3 and latitude 23 3 24 5 during the 1999 9 20 17 47 1999 9 24 08 13 period it was found that the aftershock sequence exhibited the characteristic right skewed frequency distribution and could be well described with the lognormal distribution long term memory and the possibility of scale invariance were first roughly identified through the analysis of autocorrelation and power spectrum respectively scale invariance was clearly revealed with the aid of box counting method and the box dimension was shown to be a decreasing function of the threshold magnitude level i e the weak and intense regions scaled differently to verify the existence of multifractal characteristics the aftershock sequence was transferred into a useful compact form through the multifractal formalism namely the q q and f plots the analysis confirmed the existence of multifractal characteristics in the examined aftershock sequence the origin of both the pronounced right skewness and multifractal phenomena in aftershock sequence might be interpreted in terms of the multiplicative cascade process of the stress in the earth s crust a simple two scale cantor set with unequal scales and weights was then used to fit the calculated q q plot this model fitted remarkably well the entire spectrum of scaling exponents of the examined asccme

obrechkoff methods having additional parameters for general second order differential equations abstract a class of two step implicit methods involving higher order derivatives of y for initial value problems of the form y f t y y is developed the methods involve arbitrary parameters p and q which are determined so that the methods become absolutely stable when applied to the test equation y y y 0 numerical results for bessel s and general second order differential equations are presented to illustrate that the methods are absolutely stable and are of order o h 4 o h 6 and o h 8

hochschild and cyclic homology of yang mills algebras the aim of this article is to compute the hochschild and cyclic homology groups of the yang mills algebras ym n n n 2 defined by a connes and m dubois violette in cd1 contin uing thus the study of these algebras that we have initiated in hs the computation involves the use of a spectral sequence associated to the natural filtration on the universal enveloping algebra ym n provided by a lie idealtym n inym n which is free as lie algebra 2000 mathematics subject classification 16e40 16s32 17b56 70s15 81t13

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

topological rewriting systems applied to standard bases and syntactic algebras abstract we introduce topological rewriting systems as a generalisation of abstract rewriting systems where we replace the set of terms by a topological space abstract rewriting systems correspond to topological rewriting systems for the discrete topology we introduce the topological confluence property as an approximation of the confluence property using a representation of linear topological rewriting systems with continuous reduction operators we show that the topological confluence property is characterised by lattice operations using this characterisation we show that standard bases induce topologically confluent rewriting systems on formal power series finally we investigate duality for reduction operators that we relate to series representations and syntactic algebras in particular we use duality for proving that an algebra is syntactic or not

some new implicit two step multiderivative methods for solving special second order ivp s abstract the aim of this paper is to select from the huge family of implicit two step multiderivative methods those which have better characteristics as regards the error order and error constant stability and easy of use with most of the coefficients equal to zero a few of these multiderivative methods that are a 0 stable and a stable have been obtained this feature makes them useful for solving stiff problems the stability intervals or stability regions for the methods chosen are presented some numerical tests are included the numerical test concerning the wave equation shows the good performance of the methods of lower order but a 0 stable in comparison with the poor performance of higher order methods