synchronized chaotic intermittent and spiking behavior in coupled map chains we study phase synchronization effects in a chain of nonidentical chaotic oscillators with a type i intermittent behavior two types of parameter distribution linear and random are considered the typical phenomena are the onset and existence of global all to all and cluster partial synchronization with increase of coupling increase of coupling strength can also lead to desynchronization phenomena i e global or cluster synchronization is changed into a regime where synchronization is intermittent with incoherent states then a regime of a fully incoherent nonsynchronous state spatiotemporal intermittency appears synchronization desynchronization transitions with increase of coupling are also demonstrated for a system resembling an intermittent one a chain of coupled maps replicating the spiking behavior of neurobiological networks

partial synchronization from symmetry towards stability abstract in this paper we study the existence and stability of linear invariant manifolds in a network of coupled identical dynamical systems symmetry under permutation of different units of the network is helpful to construct explicit formulae for linear invariant manifolds of the network in order to classify them and to examine their stability through lyapunov s direct method

synchronized clusters in coupled map networks ii stability analysis we study self organized and driven synchronization in some simple coupled map networks namely globally coupled networks and complete bipartite networks using both linear stability analysis and lyapunov function approach and determine stability conditions for synchronization the phase diagrams for the networks studied here have features very similar to the different kinds of structurally similar networks studied in part i lyapunov function approach shows that when any two nodes are in driven synchronization all the coupling terms in the difference between the variables of these two nodes cancel out whereas when they are in self organized synchronization the direct coupling term between the two nodes adds an extra term while the other couplings cancel out we also discuss the conditions for the occurrence of a floating node and suggest that the fluctuations of the conditional lyapunov exponent about zero can be a criterion for its occurrence

m sequences graph ideals and ladder ideals of linear type in this paper we study monomial ideals and ladder determinantal ideals of linear type and their blow up algebras our main tools are grobner bases and sagbi bases deformations and the notion of m sequence of monomials w x let r s k x be a polynomial ring over a field k equipped with a monomial order t let i be an ideal of r generated by polynomials f f consider the presentation 1 s w x w x w x c r t s r t t a r i s r f t f t 1 s i s

grobner geometry of schubert polynomials given a permutation w sn we consider a determinantal ideal iw whose generators are certain minors in the generic n n matrix filled with independent variables using emultidegrees f as simple algebraic substitutes for torus equivariant cohomology classes on vector spaces our main theorems describe for each ideal iw

cluster splitting bifurcation in a system of coupled maps abstract we consider cluster splitting bifurcations in a system of globally coupled maps as coupling parameter decreases at these transitions the number of clusters i e groups of elements with identical dynamics increases we demonstrate that different cascades of cluster splitting can occur depending on statistics of redistribution of the oscillators between new born clusters

regular and chaotic phase synchronization of coupled circle maps we study the effects of regular and chaotic phase synchronization in ensembles of coupled nonidentical circle maps cms and find phase locking regions for both types of synchronization we show that synchronization of chaotic cms is crucially influenced by the three quantities i rotation number difference ii variance of the phase evolution and iii relative duration of intervals of phase increase respect decrease in the case of regular cms only variance and rotation number difference are important it is demonstrated that with increase of noncoherence of phase evolutionsin the regular and chaotic regime the regions of the main 1 1 synchronization are usually decreased we present a chaotic synchronization in the systems of coupled nonidentical circle maps where phase entrainment occurs and it is not accompanied by bifurcations of the chaotic set for ensembles chains of coupled cms with linear and random distributions of the individual frequencies soft and hard transitions to global synchronization are found

random coupling of chaotic maps leads to spatiotemporal synchronization we investigate the spatiotemporal dynamics of a network of coupled chaotic maps with varying degrees of randomness in coupling connections while strictly nearest neighbor coupling never allows spatiotemporal synchronization in our system randomly rewiring some of those connections stabilizes entire networks at x where x is the strongly unstable fixed point solution of the local chaotic map in fact the smallest degree of randomness in spatial connections opens up a window of stability for the synchronized fixed point in coupling parameter space further the coupling b i f r at which the onset of spatiotemporal synchronization occurs scales with the fraction of rewired sites p as a power law for 0 1 p 1 is we also show that the regularizing effect of random connections can be understood from stability analysis of the probabilistic evolution equation for the system and approximate analytical expressions for the range and b i f r are obtained

singular loci of ladder determinantal varieties and schubert varieties abstract we relate certain ladder determinantal varieties associated to one sided ladders to certain schubert varieties in sl n q for a suitable n and a suitable parabolic subgroup q and we determine the singular loci of these varieties we state a conjecture on the irreducible components of the singular locus of a schubert variety in the flag variety which is a refinement of the conjecture of lakshmibai and sandhya proc indian acad sci math sci 100 1990 45 52 we prove the conjecture for a certain class of schubert varieties

f purity and rational singularity we investigate singularities which are f pure respectively f pure type a ring r of characteristic p is f pure if for every r module m 0 m r m r is exact where 1r denotes the r algebra structure induced on r via the frobenius map if r e r and s e 1r then r s rps in 1r f pure type is defined in characteristic 0 by reducing to characteristicp it is proven that when r s i is the quotient of a regular local ring s r is f pure at the prime ideal q if and only if i pi i q p here j p denotes the ideal ap i a e j several theorems result from this criterion if f is a quasihomogeneous hypersurface having weights r1 r and an isolated singularity at the origin 1 17 1ri i implies k x1 x f has f pure type at m x1 x 2 1n i ri i implies k x1 xn f does not have f pure type at m 3 1 1ri i remains unsolved but does connect with a problem that number theorists have studied for many years this theorem parallels known results about rational singularities it is also proven that classifying f pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces and that the f pure locus in the maximal spectrum of k x1 xn i where k is a perfect field of characteristic p is zariski open an important conjecture is that r fr is f pure type should imply r is f pure type whenever r is a cohen macauley normal local ring it is proven that ext1 1r r 0 is a sufficient though not necessary condition a local ring r m of characteristic p is f injective if the frobenius map induces an injection on the local cohomology modules h r h 1r an example is constructed which is f injective but not f pure from this a counterexample to the conjecture that r fr is f pure implies r is f pure is constructed however it is not a domain much less normal moreover it does not lead to a counterexample to the characteristic 0 version of the conjecture 0 introduction let r be a ring of characteristic p and let r denote the ring r viewed as an r module via the frobenius map f r rp r is f pure if for every r module m 0 r m 4 r m is exact a notion of f pure type is then defined in characteristic 0 by reduction to characteristicp f pure rings are connected with invariant theory and appear in the proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is cohen macaulay 3 it has also been demonstrated that f purity measures good singularities in the sense that it implies a great deal of simplification in the computation of local cohomology 1 received by the editors march 30 1982 and in revised form august 3 1982 1980 mathematics subject classification primary 13h99 1983 american mathematical society 0002 9947 82 0000 0829 05 25

phase ordering on small world networks with nearest neighbor edges we investigate global phase coherence in a system of coupled oscillators on small world networks constructed from a ring with nearest neighbor edges the effects of both thermal noise and quenched randomness on phase ordering are examined and compared with the global coherence in the corresponding xy model without quenched randomness it is found that in the appropriate regime phase ordering emerges at finite temperatures even for a tiny fraction of shortcuts the nature of the phase transition is also discussed

synchronization of oscillators with random nonlocal connectivity in this paper we study the existing observation in literature about synchronization of a large number of coupled maps with random nonlocal connectivity chate and manneville chaos 2 307 1992 these connectivities which lack any spatial significance can be realized in neural nets and electrical circuits it is quite interesting and of practical importance to note that a huge number of maps can be synchronized with this connectivity we show that this synchronization stems from the fact that the connectivity matrix has a finite gap in the eigenvalue spectrum in the macroscopic limit we give a quantitative explanation for the gap we compare the analytic results with the ones quoted in the above reference we also study the departures from this highly collective behavior in the low connectivity limit and show that the behavior is almost statistical for very low connectivity of late there has been considerable attention paid to the study of coupled map lattices cml in various contexts they have been used as a computationally simple and analytically tractable model for spatiotemporal systems 1 the studies on cml s have been either in one and two dimensional lattices or with global coupling in which case there is no notion of lattice geometry 2 the higher dimensional connectivities 3 or hierarchial connectivities 4 5 are studied very little one more system that has been studied is the cml with random nonlocal couplings 6 the motivation is twofold first this is an effectively high dimensional system the phenomenology in cml in higher dimensions has not been studied much and needs further investigation second there are systems like neural nets in which the local connections do not have any spatial significance there also exist systems like electrical circuits 7 in which connectivity is at one s will and such a coupling can be easily realized thus the studies of different connectivities and their effects will be useful in designing well controlled systems in this system of random nonlocal connectivity chate and manneville have presented preliminary results 6 which show that synchronization of a large number of oscillators is easily achieved with this connectivity synchronization of oscillators in spatially extended systems such as coupled oscillators is important from various points of view by achieving synchronization one can try to build huge but more controllable and better behaved systems which are effectively low dimensional 7 sometimes synchronization may serve other purposes such as sending codes that are difficult to break 8 in various contexts this problem has been subjected to several studies in the past few years 9 we would like to show that the phenomenon of synchronization can be generally understood by investigating the eigenvalue spectrum of the connectivity matrix and can help us to understand the existing observations in this paper we will explicitly illustrate how one can separate the mode leading to spatial homogeneity from the rest we will show that there exists a finite gap between the growth rates the spatially homogeneous mode and the rest in the model studied in ref 6 we will also study the departures from this behavior for lower connectivities for the linear stability analysis of the synchronized state we will study the eigenvalue spectrum of the connectivity matrix with random nonlocal connectivity we will also study the eigenspectrum of the product of such matrices we would note that a similar model of random connectivity matrix of size n3n with k nonzero elements in each row has been investigated by cook and derrida in ref 10 in connection with the random energy model the generalized random energy model and directed polymers in random media they have obtained exact analytic results for products of such matrices in the case where the matrices are sparse and the distribution function of nonzero elements is not a d function however in the model studied by chate and manneville 6 all connections have the same weight i e the distribution function of nonzero elements is a d function

f purity and rational singularity in graded complete intersection rings a simple criterion is given for determining almost completely whether the positively graded complete intersection ring r k x1 xn t g of dimension n has an f pure type singularity at m x1 x specifically if deg x a 0 for 1 0 for 1 0 the characterization given by this theorem is particularly effective if the singularity of r at m is isolated in that case 8 0 so that only the condition t 1d y lta 0 is not solved by the above result in particular it follows from work of kei ichi watanabe that if r has an isolated rational singularity then r has f pure type the converse is also almost true with the only exception being the case where t 1d e i ta 0 in proving this criterion a weak but more stable form of f purity called f contractedness is defined and explored r is f contracted in characteristic p 0 if every system of parameters for m is contracted with respect to the frobenius map f r r just as for f purity the notion of f contracted type is defined in characteristic 0 by reduction to characteristic p the two notions of f pure type and f contracted type coincide when r is gorenstein whence in particular when r is a complete intersection ring 0 introduction let r be a ring of characteristic p and let r denote the ring r viewed as an r module via the frobenius map f r rp r is f pure if for every r module m 0 r m r m is exact a notion of f pure type can then be defined in characteristic 0 by reduction to characteristic p the concept of f purity was suggested originally by the hochster roberts proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is cohen macaulay 4 the same two authors later demonstrated that f purity measures nice singularities in the sense that it implies much simplification and vanishing in the computation of local cohomology 5 the problem of identifying those rings which are f pure type is quite difficult the following special cases are known 1 the invariant ring of a reductive linear group acting on a regular ring is f pure type 5 received by the editors october 8 1985 1980 mathematics subject classification 1985 revision primary 13h99 the author wishes to thank the research council of the university of missouri for a research grant which partially supported this project 1987 american mathematical society 0002 9947 87 1 00 25 per page

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

test ideals in local rings it is shown that certain aspects of the theory of tight closure are well behaved under localization let j be the parameter test ideal for r a complete local cohen macaulay ring of positive prime characteristic for any multiplicative system u c r it is shown that ju 1r is the parameter test ideal for u 1 r this is proved by proving more general localization results for the here introduced classes of f ideals of r and f submodules of the canonical module of r which are annihilators of r modules with an action of frobenius it also follows that the parameter test ideal cannot be contained in any parameter ideal of r tight closure has produced surprising new results for instance that the absolute integral closure of a complete local domain r of prime characteristic is a cohen macaulay algebra for r as well as tremendously simple proofs for otherwise difficult theorems such as the fact that the ring of invariants of a linearly reductive group acting on a regular ring is cohen macaulay the definition of tight closure is recalled in section 1 but we refer the reader to the papers of hochster and huneke listed in the bibliography for more about its applications while primarily a prime characteristic notion tight closure offers insight into arbitrary commutative rings containing q by fairly standard reduction to characteristic p techniques despite its successes the tight closure operation which in its principal setting is a closure operation performed on ideals in a commutative noetherian ring of prime characteristic remains poorly understood for example tight closure is not known to behave well under localization given an ideal i of a ring r letting i denote the tight closure of i it is not known whether or not

phase dependent changes in local dynamic stability during walking in elderly with and without knee osteoarthritis previously we reported reduced time averaged knee local stability in the unaffected but not the affected leg of elderly with knee osteoarthritis oa compared to controls since stability may show phase related changes we reanalyzed the dataset reported previously using time dependent local stability t and also calculated time averaged local stability s for comparison we studied treadmill walking at increasing speeds focusing on sagittal plane knee movements 16 patients 12 healthy peers and 15 young subjects were measured we found a clear maximum in t i e minimum in stability at around 60 of the stride cycle stancemax t a second clear maximum swingmax t at around 95 followed by a minimum between 70 and 100 swingmin t stancemax t of both legs was significantly higher in the oa than the young control group values for healthy elderly fell between those of the other groups were significantly higher than in young adults but there was only a trend towards a significant difference with the stancemax t of the oa group s affected side time averaged and time dependent stability measures within one leg were uncorrelated while time dependent stability measures at the affected side were inversely correlated with s at the unaffected side the results indicate that time dependent local dynamic stability might provide a more detailed insight into the problems of gait stability in oa than conventional averaged local dynamic stability measures and support the notion that the paradoxical decline in unaffected side time averaged local stability may be caused by a trade off between affected and unaffected side stability

desynchronization of chaos in coupled logistic maps when identical chaotic oscillators interact a state of complete or partial synchronization may be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state remains attracting on the average considering a system of two coupled logistic maps we show that the transition to riddling will be soft or hard depending on whether the first orbit to lose its transverse stability undergoes a supercritical or subcritical bifurcation a subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state a supercritical bifurcation on the other hand is associated with the formation of a so called mixed absorbing area that stretches along the synchronized chaotic state and from which trajectories cannot escape this gives rise to locally riddled basins of attraction we present three different scenarios for the onset of riddling and for the subsequent transformations of the basins of attraction each scenario is described by following the type and location of the relevant asynchronous cycles and determining their stable and unstable invariant manifolds one scenario involves a contact bifurcation between the boundary of the basin of attraction and the absorbing area another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle produced in the riddling bifurcation and ending in a boundary crisis where the stability of an asynchronous chaotic state is destroyed finally a phase diagram is presented to illustrate the parameter values at which the various transitions occur s1063 651x 99 04509 2

coupled maps on trees we study coupled maps on a cayley tree with local nearest neighbor interactions and with a variety of boundary conditions the homogeneous state where every lattice site has the same value and the node synchronized state where sites of a given generation have the same value are both shown to occur for particular values of the parameters and coupling constants we study the stability of these states and their domains of attraction since the number of sites that become synchronized is much higher compared to that on a regular lattice control is easier to achieve a general procedure is given to deduce the eigenvalue spectrum for these states perturbations of the synchronized state lead to different spatiotemporal structures we find that a mean field like treatment is valid on this effectively infinite dimensional lattice

pattern dynamics in spatiotemporal chaos pattern selection diffusion of defect and pattern competition intermettency abstract a class of coupled map lattices is investigated as a model for the spatiotemporal chaos as the parameters are changed phase transitions among the patterns are observed pattern selection through the chaotic motion of domain is observed chaos is suppressed by the selection selection of the zigzag pattern is studied where localized defects which separate the two anti phase domains are found they change chaotically in time and move randomly in space the diffusion of defect is related with its chaotic motion collapse of zigzag pattern and the defect turbulence are studied in connection with the crisis in high dimensional dynamical systems pattern selection with longer wavelengths is found in the stronger coupling regime transition from pattern selection phase to fully developed turbulence occurs via the intermittency selective flicker like noise and the pareto distribution of lifetime of pattern are observed at the intermittency regime for the quantitative characterization of each phase pattern distributions pattern transition matrix static and dynamical entropies lifetime distribution of pattern spatiotemporal power spectra and lyapunov spectra are calculated phase transition from the pattern selection to fully developed turbulence is investigated with the use of these quantifiers where some order parameters and critical indeces are introduced

possible biomechanical origins of the long range correlations in stride intervals of walking when humans walk the time duration of each stride varies from one stride to the next these temporal fluctuations exhibit long range correlations it has been suggested that these correlations stem from higher nervous system centers in the brain that control gait cycle timing existing proposed models of this phenomenon have focused on neurophysiological mechanisms that might give rise to these long range correlations and generally ignored potential alternative mechanical explanations we hypothesized that a simple mechanical system could also generate similar long range correlations in stride times we modified a very simple passive dynamic model of bipedal walking to incorporate forward propulsion through an impulsive force applied to the trailing leg at each push off push off forces were varied from step to step by incorporating both sensory and motor noise terms that were regulated by a simple proportional feedback controller we generated 400 simulations of walking with different combinations of sensory noise motor noise and feedback gain the stride time data from each simulation were analyzed using detrended fluctuation analysis to compute a scaling exponent a this exponent quantified how each stride interval was correlated with previous and subsequent stride intervals over different time scales for different variations of the noise terms and feedback gain we obtained short range correlations 1 0 our results indicate that a simple biomechanical model of walking can generate long range correlations and thus perhaps these correlations are not a complex result of higher level neuronal control as has been previously suggested

controlling synchronous patterns in complex networks although the set of permutation symmetries of a complex network could be very large few of them give rise to stable synchronous patterns here we present a general framework and develop techniques for controlling synchronization patterns in complex network of coupled chaotic oscillators specifically according to the network permutation symmetry we design a small size and weighted network namely the control network and use it to control the large size complex network by means of pinning coupling we argue mathematically that for any of the network symmetries there always exists a critical pinning strength beyond which the unstable synchronous pattern associated to this symmetry can be stabilized the feasibility of the control method is verified by numerical simulations of both artificial and real world networks and demonstrated experimentally in systems of coupled chaotic circuits our studies show the controllability of synchronous patterns in complex networks of coupled chaotic oscillators

stochastic dynamical model for stock stock correlations we propose a model of coupled random walks for stock stock correlations the walks in the model are coupled via a mechanism that the displacement price change of each walk stock is activated by the price gradients over some underlying network we assume that the network has two underlying structures describing the correlations among the stocks of the whole market and among those within individual groups respectively each with a coupling parameter controlling the degree of correlation the model provides the interpretation of the features displayed in the distribution of the eigenvalues for the correlation matrix of real market on the level of time sequences we verify that such modeling indeed gives good fitting for the market data of us stocks

strong f regularity and generating morphisms of local cohomology modules abstract we establish a criterion for the strong f regularity of a non gorenstein cohen macaulay reduced complete local ring of dimension at least 2 and prime characteristic p we also describe an explicit generating morphism in the sense of lyubeznik for the top local cohomology module with support in certain ideals arising from an n n 1 matrix x of indeterminates for a perfect field k of characteristic p 5 these results led us to derive a simple new proof of the well known fact that the generic determinantal ring over k given by the maximal minors of x is strongly f regular

mutual synchronization and clustering in randomly coupled chaotic dynamical networks we introduce and study systems of randomly coupled maps rcm where the relevant parameter is the degree of connectivity in the system global almost synchronized states are found equivalent to the synchronization observed in globally coupled maps until a certain critical threshold for the connectivity is reached we further show that not only the average connectivity but also the architecture of the couplings is responsible for the cluster structure observed we analyse the different phases of the system and use various correlation measures in order to detect ordered non synchronized states finally it is shown that the system displays a dynamical hierarchical clustering which allows the definition of emerging graphs

synchronization and symmetry breaking bifurcations in constructive networks of coupled chaotic oscillators the spatiotemporal dynamics of networks based on a ring of coupled oscillators with regular shortcuts beyond the nearest neighbor couplings is studied by using master stability equations and numerical simulations the generic criterion for dynamic synchronization has been extended to arbitrary network topologies with zero row sum the symmetry breaking oscillation patterns that resulted from the hopf bifurcation from synchronous states are analyzed by the symmetry group theory

stability of synchronized network of chaotic electromechanical devices with nearest and all to all couplings abstract a shift invariant set of n mutually nearest neighbour or all to all coupled moving coil electromechanical devices is analytically and numerically investigated the study of the stability of synchronization process is undertaken using the master stability function msf approach the emanating properties of this method make it possible to have a general study of the network dynamics and to explain de synchronization phenomena appearing in the synchronization stability parametric areas a detailed attention is paid to the effects of the dissipative component of the dispersive dissipative coupling tested here

synchronized states in a ring of mutually coupled self sustained electrical oscillators we investigate in this paper different states of synchronization in a ring of mutually coupled self sustained electrical oscillators the good coupling parameters leading to complete and partial synchronization or disordered states are calculated using the properties of the variational equations of stability a stability map showing domains of synchronization to an external excitation locally injected in the ring is also obtained in both cases the numerical simulation validates and complements the results of the analytical investigation

synchronized clusters in coupled map networks i numerical studies we study the synchronization of coupled maps on a variety of networks including regular one and two dimensional networks scale free networks small world networks tree networks and random networks for small coupling strengths nodes show turbulent behavior but form phase synchronized clusters as coupling increases when nodes show synchronized behavior we observe two interesting phenomena first there are some nodes of the floating type that show intermittent behavior between getting attached to some clusters and evolving independently second we identify two different ways of cluster formation namely self organized clusters which have mostly intracluster couplings and driven clusters which have mostly intercluster couplings the synchronized clusters may be of dominant self organized type dominant driven type or mixed type depending on the type of network and the parameters of the dynamics we define different states of the coupled dynamics by considering the number and type of synchronized clusters for the local dynamics governed by the logistic map we study the phase diagram in the plane of the coupling constant and the logistic map parameter ensuremath mu for large coupling strengths and nonlinear coupling we find that the scale free networks and the caley tree networks lead to better cluster formation than the other types of networks with the same average connectivity for most of our study we use the number of connections of the order of the number of nodes as the number of connections increases the number of nodes forming clusters and the size of the clusters in general increase

mixed ladder determinantal varieties abstract we investigate ladder determinantal varieties defined by ideals of minors of possibly different sizes in the different regions the steps of one sided ladders l these varieties are an important generalization of the classical ladder determinantal varieties i e with equal size minors since they are very closely related to schubert varieties this being the first main result of this paper we show that they correspond to opposite cells in schubert varieties in flag varieties of type a n in consequence one deduces the normality and the cohen macaulayness of these one sided ladder determinantal varieties with ideals of mixed size minors as well as the fact that they have rational singularities next we show that up to products by affine spaces each of these varieties is a basic open set in a classical ladder determinantal variety i e with equal size minors and that it contains as a basic open set another classical ladder determinantal variety this result along with a general localisation lemma used to show it enables us to compute the divisor class group and singular locus of the coordinate rings of these varieties as well as to determine when they are gorenstein

good filtrations of symmetric algebras and strong f regularity of invariant subrings in this paper we prove that a linear action of a reductive group on a polynomial ring with good filtrations over a field of characteristic p 0 yields a strongly f regular in particular cohen macaulay invariant subring the strongly f regular property of some known examples of invariant subrings such as the coordinate rings of schubert varieties in grassmannians are recovered a similar result over a field of characteristic zero is also proved

quaternions and joint angles in an analysis of local stability of gait for different variants of walking speed and treadmill slope the authors describe an example of application of a nonlinear time series analysis directed at identifying the presence of deterministic chaos in human gait kinematic data by means of the largest lyapunov exponent lle a positive lle value is interpreted as an indicator of local instability the research was aimed at assessment of the influence of both walking speed and ground slope on the resilience of gait control to infinitesimally small perturbations that occur naturally during walking the analysis of treadmill gait data was carried out twofold 1 for the time series representing the following joint angles hip flexion extension knee flexion extension and dorsiflexion plantarflexion of the ankle and 2 for the time series representing rotations of foot tibia and femur segments through euler angles converted to a quaternion representation a comparison between both approaches as well as a dependency between treadmill inclination and lle values constitute the original aspects of this study the lle value was estimated threefold for every time series as the short term lle for both the first step and the first stride and as the long term lle over a fixed interval between the fourth and the tenth stride it was confirmed that all considered movements are characterized by positive lle values which quantify a local instability moreover a tendency to attenuate the perturbation consequences is evident in all variants of walking speed and treadmill slope

the validity of stability measures a modelling approach measures calculated from unperturbed walking patterns such as variability measures and maximum floquet multipliers are often used to study the stability of walking however it is unknown if and to what extent these measures correlate to the probability of falling we studied whether in a simple model of human walking i e a passive dynamic walker the probability of falling could be predicted from maximum floquet multipliers kinematic state variability and step time variability we used an extended version of the basic passive dynamic walker with arced feet and a hip spring the probability of falling was manipulated by varying the foot radius and hip spring stiffness or varying these factors while co varying the slope to keep step length constant the simulation data indicated that floquet multipliers and kinematic state variability correlated inconsistently with probability of falling step time variability correlated well with probability of falling but a more consistent correlation with the probability of falling was found by calculating the variability of the log transform of the step time our findings speak against the use of maximum floquet multipliers and suggest instead that variability of critical variables may be a good predictor of the probability to fall

information transfer and nontrivial collective behavior in chaotic coupled map networks the emergence of nontrivial collective behavior in networks of coupled chaotic maps is investigated by means of a nonlinear mutual prediction method the resulting prediction error is used to measure the amount of information that a local unit possesses about the collective dynamics applications to locally and globally coupled map systems are considered the prediction error exhibits phase transitions at critical values of the coupling for the onset of ordered collective behavior in these networks this information measure may be used as an order parameter for the characterization of complex behavior in extended chaotic systems

on the strength of attractors in a high dimensional system milnor attractor network robust global attraction and noise induced selection abstract strength of attractor is studied by the return rate to itself after perturbations for a multi attractor state of a globally coupled map it is found that fragile milnor attractors have a large basin volume at the partially ordered phase such dominance of fragile attractors is understood by robustness of global attraction in the phase space change of the attractor strength and basin volume against the parameter and size are studied in the partially ordered phase the dynamics is often described as milnor attractor network which leads to a new interpretation of chaotic itinerancy noise induced selection of fragile attractors is found that has a sharp dependence on the noise amplitude relevance of the observed results to neural dynamics and cell differentiation is also discussed

local estimates for entropy densities in coupled map lattices we present a method to derive an upper bound for the entropy density of coupled map lattices with local interactions from local observations to do this we use an embedding technique that is a combination of time delay and spatial embedding this embedding allows us to identify the local character of the equations of motion based on this method we present an approximate estimate of the entropy density by the correlation integral

revisiting the stability of 2d passive biped walking local behavior models of biped walking have demonstrated that stable walking motions are possible without active control stability of these motions has typically been quantified by studying the stability of an associated poincare map orbital stability however additional insight may be obtained by examining how perturbations evolve over the short term local stability for example there may be regions where small perturbations actually diverge from the unperturbed trajectory even if over the entire cycle small but perhaps not large perturbations are dissipated we present techniques to calculate local stability and demonstrate the utility of these techniques by examining the local stability of the 2d compass biped these techniques are relevant to the design of controllers to maintain stability in robots and in understanding how the neuromuscular system maintains stability in humans

straightening law and powers of determinantal ideals of hankel matrices in this paper we establish a standard monomial theory for generic hankel matrices by a generic hankel matrix we mean a matrix y yij with yij x i j 1 where the x i are indeterminates over a field k we use this structure to determine the symbolic powers and the primary decomposition of the powers of the determinantal ideals it of y further we prove that the symbolic and ordinary rees algebras of it are cohen macaulay normal domains the first standard monomial theory was developed by hodge h to study the homogeneous coordinate ring of the grassmannian variety later standard monomial theories were established for generic matrices by doubilet rota and stein drs and for generic symmetric and generic skew symmetric matrices by de concini and procesi dp these are all examples of algebras with straightening law asl for short over a poset or over a doset the abstract notion of asl was introduced and developed by eisenbud e1 and by de concini eisenbud and procesi dep2 see also bv these structures turned out to be an extremely powerful tool in studying determinantal rings and ideals arising from the above mentioned generic matrices let now x1 xn be indeterminates over an arbitrary field k for j 1 n we denote by xj the j n 1 j hankel matrix with entries x1 xn that is

clustering coding switching hierarchical ordering and control in a network of chaotic elements abstract a network of chaotic elements is investigated with the use of globally coupled maps a simple coding of many attractors with clustering is shown through the coding the attractors are organized so that their change exhibits bifurcation like phenomena a precision dependent tree is constructed which leads to the similarity of our attractor with those of spin glasses hierarchical dynamics is constructed on the tree which leads to the dynamical change of trees and the temporal change of effective degrees of freedom by a simple input on a site we can switch among attractors and tune the strength of chaos a threshold on a cluster size is found beyond which a peculiar posi nega switch occurs possible application to biological information processing is discussed with the emphasis on the fuzzy switch chaotic search and hierarchical code categorization

stability of synchronous chaos and on off intermittency in coupled map lattices in this paper we consider the stability of synchronous chaos in lattices of coupled n dimensional maps for global coupling we derive explicit conditions for computing the parameter values at which the synchronous chaotic attractor becomes unstable and bifurcates into asynchronous chaos in particular we show that after the bifurcation one generally observes on off intermittency a process in which the entire system evolves nearly synchronously but chaotically for long periods of time which are interrupted by brief bursts away from synchrony for nearest neighbor coupled systems however we show that the stability of the synchronous chaotic state is a function of the system size in particular for large systems we will not be able to observe synchronous chaos we derive a condition relating the local map s largest lyapunov exponent to the maximal system size under which one can still observe synchronous chaos and on off intermittency other issues related to the characterization of on off intermittent signals are also discussed

partial synchronization and spontaneous spatial ordering in coupled chaotic systems a model of many symmetrically and locally coupled chaotic oscillators is studied partial chaotic synchronizations associated with spontaneous spatial ordering are demonstrated very rich patterns of the system are revealed based on partial synchronization analysis the stabilities of different partially synchronous spatiotemporal structures and some dynamical behaviors of these states are discussed both numerically and analytically

hankel determinantal rings have rational singularities abstract hankel determinantal rings i e determinantal rings defined by minors of hankel matrices of indeterminates arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves they may also be viewed as linear specializations of generic determinantal rings we prove that over fields of characteristic zero hankel determinantal rings have rational singularities in the case of positive prime characteristic we prove that they are f pure independent of the characteristic we give a complete description of the divisor class groups of these rings and show that each divisor class group element is the class of a maximal cohen macaulay module

stability and transverse manifolds of periodic points for coupled maps abstract we consider the domains of stability of periodic points for linear internal and external coupled maps we obtain the approximation expressions of the transverse manifolds at periodic points which show that the transverse manifolds of periodic points are asymptotically elliptic paraboloids we point out that the action of maps on the transverse manifold has the symmetry only in two coupled maps we study in detail the hereditary properties of domains of stability of period doubling points for the internal coupling of an arbitrary one dimensional map with the help of its quadratic approximation and show that these domains follow the universal rules similar to the feigenbaum universality rules for one dimensional maps

scale free characteristics of random networks the topology of the world wide web the world wide web forms a large directed graph whose vertices are documents and edges are links pointing from one document to another here we demonstrate that despite its apparent random character the topology of this graph has a number of universal scale free characteristics we introduce a model that leads to a scale free network capturing in a minimal fashion the self organization processes governing the world wide web c 2000 elsevier science b v all rights reserved pacs 84 35 i 64 60 fr 87 23 ge

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization