transition to chaos in a shell model of turbulence abstract we study a shell model for the energy cascade in three dimensional turbulence by varying the coefficients of the non linear terms in such a way that the fundamental symmetries of navier stokes are conserved when the control parameter related to the strength of backward energy transfer is small enough the dynamical system has a stable fixed point corresponding to the kolmogorov scaling by using the bi orthogonal decomposition the transition to chaos is shown to follow the ruelle takens scenario for 0 3953 there exists a strange attractor which remains close to the kolmogorov fixed point the intermittency of the chaotic evolution and of the scaling can be described by an intermittent one dimensional map we introduce a modified shell model which has a good scaling behaviour also in the infrared region we study the multifractal properties of this model for large number of shells and for values of slightly above the chaotic transition in this case by making a local analysis of the scaling properties in the inertial range we found that the multifractal corrections seem to become weaker and weaker approaching the viscous range

on the multifractal nature of fully developed turbulence and chaotic systems it is generally argued that the energy dissipation of three dimensional turbulent flow is concentrated on a set with non integer hausdorff dimension recently in order to explain experimental data it has been proposed that this set does not possess a global dilatation invariance it can be considered to be a multifractal set the authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta model

reduced standard modules and cohomology first cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts including wiles s famous paper 1995 internal to group theory 1 cohomology plays a role in the general theory of maximal subgroups of finite groups as developed by aschbacher and scott 1985 one can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible in this case r guralnick 1986 conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups this paper provides the first general positive results on this conjecture proving that the generic 1 cohomology h 1 gen g l lim h 1 g q l see cline parshall scott and van der kallen 1977 of a finite group g q of lie type with absolutely irreducible coefficients l in the defining characteristic of g is bounded by a constant depending only on the root system in all cases we are able to improve this result to a bound on h 1 g q l itself still depending only on the root system the generic h 1 result and related results for ext 1 emerge here as a consequence of a general study of interest in its own right of the homological properties of certain rational modules red red indexed by dominant weights for a reductive group g the modules red and red arise naturally from irreducible representations of the quantum enveloping algebra u of the same type as g at a pth root of unity where p 0 is the characteristic of the defining field for g finally we apply our ext 1 bounds results of bendel nakano and pillen 2006 as well as results of sin 1993 1992 1994 on the ree and suzuki groups to obtain the non generic bounds on h 1 g q l

complexity in quantum systems we discuss the behavior of a quantum 1 2 spin coupled to a time dependent magnetic field which can be quasiperiodic or random for a quasiperiodic field the time evolution of the system on a coarse grained space has autocorrelation which does not decay and positive topological entropy the information complexity diverges as a stretched exponential although the shannon entropy is zero and there is no divergence of nearby orbits a simple random process is introduced in order to reproduce the main qualitative features of the observed weakly complex behavior

predictability in the large an extension of the concept of lyapunov exponent we investigate the predictability problem in dynamical systems with many degrees of freedom and a wide spectrum of temporal scales in particular we study the case of three dimensional turbulence at high reynolds numbers by introducing a finite size lyapunov exponent which measures the growth rate of finite size perturbations for sufficiently small perturbations this quantity coincides with the usual lyapunov exponent when the perturbation is still small compared to large scale fluctuations but large compared to fluctuations at the smallest dynamically active scales the finite size lyapunov exponent is inversely proportional to the square of the perturbation size our results are supported by numerical experiments on shell models we find that intermittency corrections do not change the scaling law of predictability we also discuss the relation between the finite size lyapunov exponent and information entropy

predictability of velocity and temperature fields in intermittent turbulence we discuss the relation between predictability and the sensitive dependence on initial conditions in turbulent flows the dependence on the reynolds number of the maximum lyapunov exponent of the flow is evaluated within the multifractal model the temporal intermittency of the degree of chaos is found to localize on the degrees of freedom corresponding to high wavenumbers we discuss the consequences for the mechanism for the growth of small perturbations we show that intermittency causes long tails in the distribution of the predictability times both in the velocity field and in a passively advected scalar field though there is no simple relation between these two times numerical evidence of this picture is provided within the framework of a cascade model for velocity and temperature fields in fully developed turbulence

periodic solutions of an infinite dimensional hamiltonian system abstract we consider the existence of periodic solutions of the unbounded hamiltonian system h t u x u h v t x u v t v x v h u t x u v for t x r unlike previous work in our case the energy functional does not satisfy the palais smale conditions

sectional rank and cohomology abstract donovan s conjecture implies a bound on the dimensions of cohomology groups in terms of the size of a sylow p subgroup and we give a proof of a stronger bound in terms of sectional p rank for dim h 1 g v we also prove a reduction theorem for higher cohomology

predictability in systems with many characteristic times the case of turbulence in chaotic dynamical systems an infinitesimal perturbation is exponentially amplified at a time rate given by the inverse of the maximum lyapunov exponent lambda in fully developed turbulence lambda grows as a power of the reynolds number this result could seem in contrast with phenomenological arguments suggesting that as a consequence of physical perturbations the predictability time is roughly given by the characteristic life time of the large scale structures and hence independent of the reynolds number we show that such a situation is present in generic systems with many degrees of freedom since the growth of a non infinitesimal perturbation is determined by cumulative effects of many different characteristic times and is unrelated to the maximum lyapunov exponent our results are illustrated in a chain of coupled maps and in a shell model for the energy cascade in turbulence

first cohomology groups of chevalley groups in cross characteristic let g be a simple chevalley group dened over fq we show that if r does not divide q and k is an algebraically closed eld of characteristic r then very few irreducible kg modules have nonzero h 1 g v we also give an explicit upper bound for dimh 1 g v for v an irreducible kg module that does not depend on q but only on the rank of the group cline parshall and scott showed that such a bound exists when rjq we obtain extremely strong bounds in the case that a borel subgroup has no xed points on v

multiscaling in multifractals multiscaling is shown to be a consequence of multifractality when a lower cutoff is introduced in calculations of correlation functions after a suitable rescaling the correlation function data for different values of seem to fall onto a single curve in the multiscaling regime however we show that there is not a unique functional form at varying but a spread very close to a single curve for each this curve can be computed analytically in terms of the f a spectrum which characterizes the multifractal part of this spectrum can thus be obtained by computing only one moment of the weights at 0

on the number of simple modules in a block of a finite group abstract we prove that if b is a p block with non trivial defect group d of a finite p solvable group g then l b p r where r is the sectional rank of d we remark that there are infinitely many p blocks b with non abelian defect groups and l b p r 1 we conjecture that the inequality l b p r holds for an arbitrary p block with defect group of sectional rank r we show this to hold for a large class of p blocks of various families of quasi simple and nearly simple groups

anomalous diffusion as modeled by a nonstationary extension of brownian motion if the mean square displacement of a stochastic process is proportional to t ensuremath beta ensuremath beta ensuremath ne 1 then it is said to be anomalous we construct a family of markovian stochastic processes with independent nonstationary increments and arbitrary but a priori specified mean square displacement we label the family as an extended brownian motion and show that they satisfy a langevin equation with time dependent diffusion coefficient if the time derivative of the variance of the process is homogeneous then by computing the fractal dimension it can be shown that the complexity of the family is the same as that of the brownian motion for two particles initially separated by a distance x the finite size lyapunov exponent fsle measures the average rate of exponential separation to a distance ax an analytical expression is developed for the fsles of the extended brownian processes and numerical examples presented the explicit construction of these processes illustrates that contrary to what has been stated in the literature a power law mean square displacement is not necessarily related to a breakdown in the classical central limit theorem clt caused by for example correlation fractional brownian motion or correlated continuous time random walk schemes or infinite variance levy motion the classical clt coupled with nonstationary increments can and often does give rise to power law moments such as the mean square displacement

on donovan s conjecture abstract this article contains reduction theorems for some weaker variants of donovan s conjecture which supposes that for every finite group d of prime order p there are only finitely many morita equivalence classes of p blocks of finite groups having defect groups isomorphic to d i when restricting the conjecture to the case of abelian defect groups it can be reduced to only considering p blocks of a class of groups that are close to being direct products of simple groups ii when restricting the conjecture to the case of principal blocks of finite groups having an abelian sylow p subgroup it can be reduced to only considering p blocks of simple groups a weaker version of donovan s conjecture would suppose that there are only finitely many cartan matrices of p blocks of finite groups with defect group isomorphic to a given one iii this conjecture can be reduced to only considering p blocks of quasisimple groups with centers having orders prime to p

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

normalizers of primitive permutation groups abstract let g be a transitive normal subgroup of a permutation group a of finite degree n the factor group a g can be considered as a certain galois group and one would like to bound its size one of the results of the paper is that a g n if g is primitive unless n 3 4 54 38 58 or 316 this bound is sharp when n is prime in fact when g is primitive out g n unless g is a member of a given infinite sequence of primitive groups and n is different from the previously listed integers many other results of this flavor are established not only for permutation groups but also for linear groups and galois groups

predictability in spatially extended systems the predictability problem is discussed in turbulent fluids and in spatially extended systems we perform a numerical analysis of the spatial propagation of a small perturbation initially localized at a point of a one dimensional lattice of coupled maps where the interactions can be either local or non local in space in both cases the nonlinear terms in the dynamics of the perturbation cannot be neglected and the predictability problem cannot be reduced to the linearized evolution equations only for non local interactions is the predictability time proportional to the inverse lyapunov exponent lambda as in dynamical systems with a few degrees of freedom for local interactions it is proportional to the size of the system and is practically independent of the chaoticity degree lambda

slow and fast dynamics in coupled systems a time series analysis view abstract we study the dynamics of systems with different timescales when access only to the slow variables is allowed we use the concept of finite size lyapunov exponent fsle and consider both the cases when the equations of motion for the slow components are known and the situation when a scalar time series of one of the slow variables has been measured a discussion on the effects of parametrizing the fast dynamics is given we show that although the computation of the largest lyapunov exponent can be practically infeasible in complex dynamical systems the computation of the fsle allows to extract information on the characteristic time and on the predictability of the large scale slow time dynamics even with moderate statistics and unresolved small scales