model reduction via parametrized locally invariant manifolds some examples a method for model reduction in non linear ode systems is demonstrated through computational examples the method does not require an implicit separation of time scales in the fine dynamics to be effective from the computational standpoint the method has the potential of serving as a subgrid modeling tool from the physical standpoint it provides a model for interpreting and describing history dependence in coarse grained response of an autonomous system

erratic behavior of invariant circles in standard like mappings abstract we study mappings like the standard map but with the sine replaced by another function by varying a parameter in this function it changes smoothly from the sine to a piecewise linear pwl function the function is always analytic except when it is pwl taking the function close enough to the pwl function it is found that kam tori can reappear after they disappeared when the nonlinearity is increased when plotting the boundary around the area where a kam torus with noble winding number exists in the plane of the non linearity versus the distance of the function to the pwl function self similarity is observed

existence of dynamical hull functions with two variables for the ac driven frenkel kontorova model abstract in this paper we are concerned with the dynamics of incommensurate structures in the overdamped limit case of the frenkel kontorova model with periodic driving force we show in the infinite dimensional phase space there exists an ordered circle on which both the poincare map and the space shift map induce orientation preserving circle homeomorphisms with rotation numbers and respectively the latter characterizing the mean spacing of particles furthermore we prove that the average velocity for the particle chain with irrational exists and t where t denotes the period of the driving force and it is continuous and nondecreasing with respect to f the average of the external driving force if the shift map on the invariant ordered circle is ergodic we demonstrate the existence of two variable dynamical hull functions and that there are no mode locking behaviors

critical behavior of a kam surface i empirical results kolmogorov arnol d moser kam surfaces are studied in the context of a perturbed two dimensional twist map in particular we ask how a kam surface can disappear as the perturbation parameter is increased following greene we use cycles to numerically construct the kam curve and discover that at the critical coupling it shows structure at all length scales aspects of this structure are fitted by a scaling analysis critical indices and scaling functions are determined numerically some evidence is presented which suggests that the results are universal

frenkel kontorova model with toda interactions we have studied the frenkel kontorova fk model with toda interactions the phase diagram is found to be asymmetric due to the exponential form of the toda interaction the reflection symmetry observed in the standard fk model is broken here the singularity spectrum and the generalized dimension are calculated and their dependence on the nonlinearity parameter is discussed the critical exponents of the gap in the phonon spectrum the correlation length and the peierls nabarro barrier are found to be the same as those in the standard fk model and are independent of the nonlinearity parameter

equivalent classes of critical circles we present numerical evidence that the fractal properties of the critical invariant circles of a typical area preserving twist map as summarized by the spectrum and the generalized dimensions d q depend only on the tails in the continued fraction expansion of the corresponding rotation numbers and d q are numerically the same for all critical invariant circles of the standard and sine maps which have the rotation numbers with the same periodic tail

quasiperiodic dynamics in hamiltonian 1 1 2 degree of freedom systems far from integrability abstract the subject of this paper is two quasiperiodicity in a large class of one and a half degree of freedom hamiltonian systems the main result is that such systems have invariant tori for any internal frequency that is of constant type and sufficiently large relative to the forcing frequency an explicit bound on the minimum value of the internal frequency is presented the systems under consideration are not required to be small perturbations of integrable ones

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

nonanalytic twist maps and frenkel kontorova models we give a review of our recent works on a class of twist maps and frenkel kontorova fk models in which the external potential function is nonanalytic c1 and endowed with a variable degree of inflection z when z 3 recurrence of invariant curves has been observed an inverse residue criterion for the reappearance of an invariant curve complementary to the residue criterion for the disappearance of an invariant curve is introduced to make a precise determination of the reappearance point we have also studied the local and global scaling behaviors of these curves the critical exponents the singularity spectrum and the generalized dimensions are found to vary with z when 2 z 3 but remain the same for z 3 the degree of inflection thus plays a role reminiscent of that of dimensionality in phase transitions with z 2 and 3 corresponding respectively to the lower and upper critical dimensions when the same nonanalytic function is substituted for the sinusoidal external potential in the standard fk model the same conclusions are reached there exists a sequence of pinning depinning transitions and the critical exponents of the phonon gap the coherence length and the peierls nabarro barrier all show the same dependence on z

fractal boundary for the existence of invariant circles for area preserving maps observations and renormalisation explanation abstract breakup of a golden invariant circle is studied in an extended version of the standard map with two parameters the critical line in parameter space turns out to have a cantor set of cusps which can be organized in a self similar tree a theoretical explanation is conjectured in terms of a horseshoe for a renormalisation operator the results of some other researchers on similar systems are shown to fit this explanation as well

a roughness measure for fuzzy sets recently an attempt of integration between the theories of fuzzy sets and rough sets has resulted in providing a roughness measure for fuzzy sets m banerjee s k pal roughness of a fuzzy set information sciences 93 1996 235 246 essentially banerjee and pal s roughness measure depends on parameters that are designed as thresholds of definiteness and possibility in membership of the objects to a fuzzy set in this paper we first remark that this measure of roughness has several undesirable properties and then propose a parameter free roughness measure for fuzzy sets based on the notion of the mass assignment of a fuzzy set several interesting properties of this new measure are examined furthermore we also discuss how the proposed approach is used to describe the rough approximation quality of a fuzzy classification

hyperbolic cantori have dimension zero it is proved that cantori for area preserving twist maps have dimension zero only finitely many orbits of gaps and gap widths go to zero exponentially in each orbit if they are hyperbolic

roughness in rings in this paper basic notions of the rough set theory will be given in fact the paper concerns a relationship between rough sets and ring theory we shall introduce the notion of rough subring resp ideal with respect to an ideal of a ring which is an extended notion of a subring resp ideal in a ring and we shall give some properties of the lower and the upper approximations in a ring

homogenization of fully overdamped frenkel kontorova models in this paper we consider the fully overdamped frenkel kontorova model this is an infinite system of coupled first order odes each ode represents the microscopic evolution of one particle interacting with its neighbors and submitted to a fixed periodic potential after a proper rescaling a macroscopic model describing the evolution of densities of particles is obtained we get this homogenization result for a general class of frenkel kontorova models the proof is based on the construction of suitable hull functions in the framework of viscosity solutions

rough set approach to multi attribute decision analysis abstract we review the methodology of the rough set analysis of multi attribute decision problems rough set theory proved to be a useful tool for analysis of a vague description of decision situations it answers two basic questions related to multi attribute decision problems one about explanation of a decision situation and another about prescription of some decisions basing on analysis of a decision situation we define four classes of multi attribute decision problems depending on the structure of their representation its interpretation and the kind of questions related then we characterize the rough set methodology for each particular class of decision problems we use simple practical examples to illustrate this presentation a review of related literature is made throughout the paper

optical hamiltonians and symplectic twist maps abstract this paper concentrates on optical hamiltonian systems of t t n i e those for which hpp is a positive definite matrix and their relationship with symplectic twist maps we present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems the novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system

one dimensional infinite horizon variational problems arising in continuum mechanics in this paper we study a variational problem for real valued functions defined on an infinite semiaxis of the line to wit given x 2 we seek a minimal solution to the problem minimize the functional given by i left w left cdot right right int limits 0 infty f left w left s right ddot w left s right right ds w in ax left upsilon in w loc 2 1 left 0 infty right left upsilon left 0 right dot upsilon left 0 right x right right p

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

exact results for an approximate renormalisation scheme and some predictions for the breakup of invariant ori abstract an approximate renormalisation scheme is derived for the breakup of invariant tori of arbitrary winding ratio in hamiltonian systems of one and a half degrees of freedom similar to that of escande and doveil it is a free semi group with two generators this scheme is solved exactly for its orbits stable manifolds unstable manifolds and critical set various results are found including a cantor set of universal fractal diagrams the robustness of noble tori and a scaling law for areas near critical circles

rotation entropy and equilibrium states for a dynamical system x t and function f x rd we consider the corresponding generalised rotation set this is the convex subset of rd consisting of all integrals of f with respect to t invariant probability measures we study the entropy h e of rotation vectors e and relate this to the directional entropy lhl l of geller misiurewicz for x t a mixing subshift of finite type and f of summable variation we prove that if the rotation set is strictly convex then the functions h and h are in fact one and the same for those rotation sets which are not strictly convex we prove that 1h e and h e can differ only at non exposed boundary points e

evolution of worldwide stock markets correlation structure and correlation based graphs we investigate the daily correlation present among market indices of stock exchanges located all over the world in the time period january 1996 to july 2009 we discover that the correlation among market indices presents both a fast and a slow dynamics the slow dynamics reflects the development and consolidation of globalization the fast dynamics is associated with critical events that originate in a specific country or region of the world and rapidly affect the global system we provide evidence that the short term time scale of correlation among market indices is less than 3 trading months about 60 trading days the average values of the nondiagonal elements of the correlation matrix correlation based graphs and the spectral properties of the largest eigenvalues and eigenvectors of the correlation matrix are carrying information about the fast and slow dynamics of the correlation of market indices we introduce a measure of mutual information based on link co occurrence in networks in order to detect the fast dynamics of successive changes of correlation based graphs in a quantitative way

minimum spanning tree filtering of correlations for varying time scales and size of fluctuations based on a recently proposed q dependent detrended cross correlation coefficient ensuremath rho q j kwapie ifmmode acute n else n fi p o ifmmode acute s else s fi wi ifmmode mbox k e else k e fi cimka and s dro ifmmode dot z else z fi d ifmmode dot z else z fi phys rev e 92 052815 2015 we generalize the concept of the minimum spanning tree mst by introducing a family of q dependent minimum spanning trees q mathrm mst mathrm s that are selective to cross correlations between different fluctuation amplitudes and different time scales of multivariate data they inherit this ability directly from the coefficients ensuremath rho q which are processed here to construct a distance matrix being the input to the mst constructing kruskal s algorithm the conventional mst with detrending corresponds in this context to q 2 in order to illustrate their performance we apply the q mathrm msts to sample empirical data from the american stock market and discuss the results we show that the q mathrm mst graphs can complement ensuremath rho q in disentangling hidden correlations that cannot be observed in the mst graphs based on ensuremath rho mathrm dcca and therefore they can be useful in many areas where the multivariate cross correlations are of interest as an example we apply this method to empirical data from the stock market and show that by constructing the q mathrm msts for a spectrum of q values we obtain more information about the correlation structure of the data than by using q 2 only more specifically we show that two sets of signals that differ from each other statistically can give comparable trees for q 2 while only by using the trees for q ensuremath ne 2 do we become able to distinguish between these sets we also show that a family of q mathrm msts for a range of q expresses the diversity of correlations in a manner resembling the multifractal analysis where one computes a spectrum of the generalized fractal dimensions the generalized hurst exponents or the multifractal singularity spectra the more diverse the correlations are the more variable the tree topology is for different q s as regards the correlation structure of the stock market our analysis exhibits that the stocks belonging to the same or similar industrial sectors are correlated via the fluctuations of moderate amplitudes while the largest fluctuations often happen to synchronize in those stocks that do not necessarily belong to the same industry

topological properties of stock networks based on minimal spanning tree and random matrix theory in financial time series abstract we investigated the topological properties of stock networks constructed by a minimal spanning tree we compared the original stock network with the estimated network the original network is obtained by the actual stock returns while the estimated network is the correlation matrix created by random matrix theory we found that the consistency between the two networks increases as more eigenvalues are considered in addition we suggested that the largest eigenvalue has a significant influence on the formation of stock networks

dynamic asset trees and black monday abstract the minimum spanning tree based on the concept of ultrametricity is constructed from the correlation matrix of stock returns the dynamics of this asset tree can be characterised by its normalised length and the mean occupation layer as measured from an appropriately chosen centre called the central node we show how the tree length shrinks during a stock market crisis black monday in this case and how a strong reconfiguration takes place resulting in topological shrinking of the tree

aubry mather sets and birkhoff s theorem for geodesic flows on the two dimensional torus in this paper we state the graph property for incompressible continuouse tori invariant under goedesic flows of riemannian metrics on the two dimensional torus also our method gives a new proof of birkhoff s theorem for twist maps of the cylinder we prove that if there exist an invariant incompressible torus of geodesic flow with irrational rotation number then it necessarily contains the aubry mather set with this rotation number

comparing minimum spanning trees of the italian stock market using returns and volumes abstract we have built the network of the top 100 italian quoted companies in the decade 2001 2011 using four different methods comparing the resulting minimum spanning trees for methods and industry sectors our starting method is based on person s correlation of log returns used by several other authors in the last decade the second one is based on the correlation of symbolized log returns the third of log returns and traded money and the fourth one uses a combination of log returns with traded money we show that some sectors correspond to the network s clusters while others are scattered in particular the trading and apparel sectors we analyze the different graph s measures for the four methods showing that the introduction of volumes induces larger distances and more homogeneous trees without big clusters

strange non chaotic attractor in a quasiperiodically forced circle map abstract we show that in the quasiperiodically forced circle map strange non chaotic attractors can appear for non linearities far from the border of chaos the destruction of a two frequency quasiperiodic torus connected with the appearance of a strange non chaotic attractor is described in detail this strange non chaotic attractor is characterized by logarithmically slow diffusion of the phase it is shown that in this regime the high order phase locking states disappear and the rotation number varies rather smoothly with the parameters

multidimensional minimal spanning tree the dow jones case abstract this paper introduces a new methodology in order to construct minimal spanning trees mst and hierarchical trees ht using the information provided by more than one variable in fact the symbolic time series analysis stsa approach is applied to the dow jones companies using information not only from asset returns but also for trading volume the us stock market structure is obtained showing eight clusters of companies and general electric as a central node in the tree we use different partitions showing that the results do not depend on the particular partition in addition we apply monte carlo simulations suggesting that the tree is not the result of random connections

topology of the correlation networks among major currencies using hierarchical structure methods abstract we studied the topology of correlation networks among 34 major currencies using the concept of a minimal spanning tree and hierarchical tree for the full years of 2007 2008 when major economic turbulence occurred we used the usd us dollar and the tl turkish lira as numeraires in which the usd was the major currency and the tl was the minor currency we derived a hierarchical organization and constructed minimal spanning trees msts and hierarchical trees hts for the full years of 2007 2008 and for the 2007 2008 period we performed a technique to associate a value of reliability to the links of msts and hts by using bootstrap replicas of data we also used the average linkage cluster analysis for obtaining the hierarchical trees in the case of the tl as the numeraire these trees are useful tools for understanding and detecting the global structure taxonomy and hierarchy in financial data we illustrated how the minimal spanning trees and their related hierarchical trees developed over a period of time from these trees we identified different clusters of currencies according to their proximity and economic ties the clustered structure of the currencies and the key currency in each cluster were obtained and we found that the clusters matched nicely with the geographical regions of corresponding countries in the world such as asia or europe as expected the key currencies were generally those showing major economic activity

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

recurrence of kolmogorov arnold moser tori in nonanalytic twist maps we study a class of twist maps where the functiong 1 2 z 1 is nonanalytic c1 and endowed with a varying degree of inflectionz whenz 3 reappearance of a kam torus after its breakup has been observed we introduce an inverse residue criterion to determine the reappearance point scaling behavior at the transition points is also studied for 2 z 3 the scaling exponents are found to vary withz whereas forz 3 they are independent ofz in this sensez 3 plays a role quite similar to that of the upper critical dimension in phase transitions

destruction of invariant tori in pendulum type equations abstract in this paper we prove that if there exists an invariant torus with the rotation number 1 in the pendulum type equation x q 0 x t x for a given potential q 0 q 0 t x c t 2 and is a liouville number then for any neighborhood n q 0 of q 0 in the c topology there exists a potential q q t x n q 0 such that the system x q x t x does not admit any invariant torus with the rotation number 1 this confirms j moser s suggestion in bol soc brasil mat 20 1981 29 45

an axiomatic characterization of a fuzzy generalization of rough sets in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties several authors have proposed various fuzzy generalizations of rough approximations in this paper we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication then we find the assumptions which permit a given fuzzy set theoretic operator to represent a upper or lower approximation derived from a special fuzzy relation different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations and different sets of axioms of fuzzy set theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator finally we study the composition of two approximation spaces it is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces

generalized frenkel kontorova models we have studied two generalized frenkel kontorova fk models in one of them the interatomic potential is changed from simple harmonic to toda in the other the external potential is changed from sinusoidal to nonanalytic their critical behaviors are found to be markedly different in the toda case the local critical exponents are the same as those in the standard fk model however the global singularity spectrum and the generalized dimension depend on the anharmonicity parameter in the nonanalytic fk model the critical exponents are found to vary with the degree of inflection z for 2 z 3 a sequence of pinning and depinning transitions reflecting the recurrence of kolmogorov arnol d moser tori has also been observed

pruning a minimum spanning tree abstract this work employs various techniques in order to filter random noise from the information provided by minimum spanning trees obtained from the correlation matrices of international stock market indices prior to and during times of crisis the first technique establishes a threshold above which connections are considered affected by noise based on the study of random networks with the same probability density distribution of the original data the second technique is to judge the strength of a connection by its survival rate which is the amount of time a connection between two stock market indices endures the idea is that true connections will survive for longer periods of time and that random connections will not that information is then combined with the information obtained from the first technique in order to create a smaller network in which most of the connections are either strong or enduring in time

on the influence of the kinetic energy on the stability of equilibria of natural lagrangian systems natural lagrangian systems t on r2 described by the equation frac d dt frac partial t partial dot q frac partial t partial q frac partial pi partial q are considered where t q q is a positive definite quadratic form in q and q has a critical point at 0 it is constructively proved that there exist a c potential energy and two c kinetic energies t and t such that the equilibrium q t 0 is stable for the system t and unstable for the system t equivalently it is established that for c natural systems the kinetic energy can influence the stability in the analytic category this is not true

emergence of time horizon invariant correlation structure in financial returns by subtraction of the market mode we investigate the emergence of a structure in the correlation matrix of assets returns as the time horizon over which returns are computed increases from the minutes to the daily scale we analyze data from different stock markets new york paris london milano and with different methods in addition to the usual correlations we also analyze those obtained by subtracting the dynamics of the center of mass i e the market mode we find that when the center of mass is not removed the structure emerges as the time horizon increases from splitting a single large cluster into smaller ones by contrast when the market mode is removed the structure of correlations observed at the daily scale is already well defined at very high frequency 5 min in the new york stock exchange moreover this structure accounts for 80 of the classification of stocks in economic sectors similar results though less sharp are found for the other markets we also find that the structure of correlations in the overnight returns is markedly different from that of intraday activity

soliton energies in the standard map beyond the chaotic threshold the birkhoff moser theorem guarantees the existence of an analytical reduction to its normal form of an area preserving map in the neighbourhood of an unstable fixed point the stable and unstable manifolds emanating from this point are then simply the images of the axes in the normal coordinates for sufficiently high parameter values of the standard map the series for the manifolds can be resummed into a closed form providing good approximations to the first loops of the manifolds the authors thus obtain simple approximations for a pair of homoclinic orbits for parameters where most orbits are chaotic the frenkel kontorova model consists of an infinite sequence of equal springs and masses under the action of a periodic potential the configuration where all the masses lie at the minima of the potential corresponds to the unstable fixed point of the standard map a soliton or discommensuration where one of the minima is missed is represented by a homoclinic orbit using the normal form they calculate the soliton energy and its pinning energy good agreement is found with the pinning energy of pokrovsky 1981 obtained as a perturbation from the continuum approximation

percival variational principle for invariant measures and commensurate incommensurate phase transitions in one dimensional chains we prove for a one dimensional system of classical particles with potential energy u alpha gamma sum limits n left alpha v x n f x n 1 x n gamma right the existence of such a smooth function 0 0 that the system with potential energyu has the equilibrium state at the temperaturet 0 this is the incommensurate phase with the ratio of periods equal to the prescribed irrational number badly approximated by rational ones a simple geometric condition for the invariant curve of the corresponding dynamical system is established under which it is the support of the invariant measure minimizing percival s energy functional

singularity spectrum of a critical kam torus the authors numerically determine the f alpha spectrum of scaling indices for critical kam tori in the area preserving standard map similarities and differences between the corresponding orbits of circle maps are discussed

spatial chaos in weakly dispersive and viscous media a nonperturbative theory of the driven kdv burgers equation abstract the asymptotic travelling wave solution of the kdv burgers equation driven by the long scale periodic driver is constructed the solution represents a shock train in which the quasi periodic sequence of dispersive shocks or soliton chains is interspersed by smoothly varying regions it is shown that the periodic solution which has the spatial driver period undergoes period doublings as the governing parameter changes two types of chaotic behavior are considered the first type is a weak chaos where only a small chaotic deviation from the periodic solution occurs the second type corresponds to the developed chaos where the solution ignores the driver period and represents a random sequence of uncorrelated shocks in the case of weak chaos the shock coordinate being repeatedly mapped over the driver period moves on a chaotic attractor while in the case of developed chaos it moves on a repeller both solutions depend on a parameter indicating the reference shock position in the shock train the structure of a one dimensional set to which this parameter belongs is investigated this set contains measure one intervals around the fixed points in the case of periodic or weakly chaotic solutions and it becomes a fractal in the case of strong chaos the capacity dimension of this set is calculated

amount of rotation about a point and the morse index for the case of exact area preserving monotone twist diffeomorphisms we give formulas relating the amount of rotation about an orbit and certain morse indices

correlation and network analysis of global financial indices random matrix theory rmt and network methods are applied to investigate the correlation and network properties of 20 financial indices the results are compared before and during the financial crisis of 2008 in the rmt method the components of eigenvectors corresponding to the second largest eigenvalue form two clusters of indices in the positive and negative directions the components of these two clusters switch in opposite directions during the crisis the network analysis uses the fruchterman reingold layout to find clusters in the network of indices at different thresholds at a threshold of 0 6 before the crisis financial indices corresponding to the americas europe and asia pacific form separate clusters on the other hand during the crisis at the same threshold the american and european indices combine together to form a strongly linked cluster while the asia pacific indices form a separate weakly linked cluster if the value of the threshold is further increased to 0 9 then the european indices france germany and the united kingdom are found to be the most tightly linked indices the structure of the minimum spanning tree of financial indices is more starlike before the crisis and it changes to become more chainlike during the crisis the average linkage hierarchical clustering algorithm is used to find a clearer cluster structure in the network of financial indices the cophenetic correlation coefficients are calculated and found to increase significantly which indicates that the hierarchy increases during the financial crisis these results show that there is substantial change in the structure of the organization of financial indices during a financial crisis

angular dynamical spectra a new method for determining frequencies weak chaos and cantori we define the invariant spectra of rotation angles and twist angles angular dynamical spectra and study the properties of their moments angular moments in a model two dimensional map the angular moments give the main frequencies of the orbits a main frequency is defined both for a regular and chaotic orbit for kam curves around a centre this frequency corresponds to the rotation number inside islands of stability we obtain both the main frequency rotation number and the epicyclic frequency a fast detection of thin chaotic layers is obtained on the basis of the behaviour of the frequency curve we explore the resonant structure near a last kam boundary the secondary islands of various multiplicities forming farey sequences and the noble tori between them are located we find a criterion to determine whether various resonant chaotic zones communicate or are separated by invariant tori

rough variables with values in measurable spaces an x valued rough variable is introduced as a measurable function from a rough space to a measurable space x based on the concept of x valued rough variables some properties of x valued rough variables are studied furthermore the concept of convergence is presented and convergence of x valued rough variables is discussed

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

aubry mather theory and hamilton jacobi equations we review and extend some results of jauslin kreiss moser and weinan e on the smooth approximation of mather sets

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

a rst a generalization of rough set theory the paper presents a transition from the crisp rough set theory to a fuzzy one called alpha rough set theory or in short a rst all basic concepts or rough set theory are extended i e information system indiscernibility dependency reduction core definability approximations and boundary the resulted theory takes into account fuzzy data and allows the approximation of fuzzy concepts besides the control of knowledge granularity is natural in a rst which is based on a parameterized indiscernibility relation a rst is developed to recognize non deterministic relationships using notions as a dependency a reduct and so forth on the other hand we introduce a notion of relative dependency as an alternative of the absolute definibility presented in rough set theory the extension a rst leads naturally to the new concept of alpha rough sets which represents sets with fuzzy non empty boundaries 2000 elsevier science inc all rights reserved

modular smoothing of action the central quantity in the theory of transport for hamiltonian systems and in particular the area preserving twist maps is the action of rotational periodic orbits usually this is a complicated discontinuous function of two arguments some perturbation parameter k and a rational rotation number denoted by we applied the idea of modular smoothing to this complicated fractal our main result is that all the information contained in the fractal can be retrieved from a set of continuous or smooth functions of one variable

rough sets and boolean reasoning abstract in this article we discuss methods based on the combination of rough sets and boolean reasoning with applications in pattern recognition machine learning data mining and conflict analysis

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

degree stability of a minimum spanning tree of price return and volatility we investigate the time series of the degree of minimum spanning trees msts obtained by using a correlation based clustering procedure which starts from i asset return and ii volatility time series the mst is obtained at different times by computing correlation among time series over a time window of fixed length t we find that the mst of asset return is characterized by stock degree values which are more stable in time than the ones obtained by analyzing a mst computed starting from volatility time series our analysis also shows that the degree of stocks has a very slow dynamics with a time scale of several years in both cases

local normalization uncovering correlations in non stationary financial time series abstract the measurement of correlations between financial time series is of vital importance for risk management in this paper we address an estimation error that stems from the non stationarity of the time series we put forward a method to rid the time series of local trends and variable volatility while preserving cross correlations we test this method in a monte carlo simulation and apply it to empirical data for the s p 500 stocks

correlations in commodity markets abstract in this paper we analyzed dependencies in commodity markets investigating correlations of future contracts for commodities over the period 1998 09 01 2007 12 14 we constructed a minimal spanning tree based on the correlation matrix the tree provides evidence for sector clusterization of investigated contracts we also studied dynamical properties of commodity dependencies it turned out that the market was constantly getting more correlated within the investigated period although the increase of correlation was distributed non uniformly among all contracts and depended on contracts branches

dynamics of market correlations taxonomy and portfolio analysis the time dependence of the recently introduced minimum spanning tree description of correlations between stocks called the asset tree has been studied in order to reflect the financial market taxonomy the nodes of the tree are identified with stocks and the distance between them is a unique function of the corresponding element of the correlation matrix by using the concept of a central vertex chosen as the most strongly connected node of the tree an important characteristic is defined by the mean occupation layer during crashes due to the strong global correlation in the market the tree shrinks topologically and this is shown by a low value of the mean occupation layer the tree seems to have a scale free structure where the scaling exponent of the degree distribution is different for business as usual and crash periods the basic structure of the tree topology is very robust with respect to time we also point out that the diversification aspect of portfolio optimization results in the fact that the assets of the classic markowitz portfolio are always located on the outer leaves of the tree technical aspects such as the window size dependence of the investigated quantities are also discussed

interface interactions in modulated phases and upsilon points certain features in frenkel kontorova and other models of phases with a one dimensional modulation can be analyzed by assuming parallel interfaces separating sets of lattice planes belonging to two different phases and treating the free energy to create interfaces as well as the interaction of two three or more interfaces as phenomenological parameters a strategy employed by fisher and szpilka for interacting defects can be extended to the case of interfaces allowing a systematic study of the phase diagram by ignoring all interface interactions and then successively taking into account pair triple and higher order terms the possible phase diagrams which can occur near the point where 0 include various sorts of endpoints analogous to critical endpoints an accumulation point of first order transitions and triple points and a self similar structure which we call an upsilon point which turns out to be an accumulation point of an infinite number of segments of first order transition lines each of which terminates in two upsilon points

scaling exponents at the transition by breaking of analyticity for incommensurate structures abstract a wide class of solid state models support incommensurate structures they are mostly given by either an analytic function in which case they are free to slide or a discontinuous function in which case there is a non zero minimum force required for depinning and a non zero minimum phonon frequency i propose that the boundary between these two types of state is given at least in part by the stable manifold of a fixed point of a renormalisation operator this permits one to perdict scaling laws for the depinning force phonon gap elasticity effective mass and other quantities

to lag or not to lag how to compare indices of stock markets that operate on different times abstract financial markets worldwide do not have the same working hours as a consequence the study of correlation or causality between financial market indices becomes dependent on whether we should use all indices on the same day or lagged indices in computations of correlation matrices the answer this article proposes is that we should consider both by representing original and lagged indices in the same network we then obtain a better understanding of how indices that operate on different hours relate to each other we use a diverse range of 79 stock market indices from around the world and study their correlation structure the eigenvalues and eigenvectors of their correlations under different time periods and volatility as well as the differences between the working hours of the stock exchanges in order to analyze the possible time zone effects and suggest ways to remove them we also analyze the enlarged correlation matrix obtained from original and lagged indices and examine a network structure derived from it thus showing connections between lagged and original indices that could not be well represented before

lyapunov exponents of minimizing measures for globally positive diffeomorphisms in all dimensions the globally positive diffeomorphisms of the 2n dimensional annulus are important because they represent what happens close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite for these globally positive diffeomorphisms an aubry mather theory was developed by garibaldi and thieullen that provides the existence of some minimizing measures using the two green bundles g and g that can be defined along the support of these minimizing measures we will prove that there is a deep link between the angle between g and g along the support of the considered measure mu the size of the smallest positive lyapunov exponent of mu the tangent cone to the support of mu

recent developments in the theory of hamiltonian systems area preserving mappings of an annulus occur as poincare mappings of hamiltonian systems they were studied extensively by g d birkhoff recently aubry and mather investigated the subclass of so called monotone twist mappings for which they constructed independently closed invariant cantor sets their work led to important new results their theory is discussed and related to hamiltonian systems satisfying a legendre condition the connection of this theory with the stability problem kam theory and in particular the disintegration of invariant tori is discussed various constructions of mather sets are explained in which minimal solutions of variational problems play a central role this theory has a close relation to the differential geometric investigations by morse and hedlund on geodesics on two dimensional surfaces

universal scaling of critical quasiperiodic orbits in a class of twist maps recently we have shown that the fractal properties of the critical invariant circles of the standard map as summarized by the spectrum and the generalized dimensions d q depend only on the tails in the continued fraction expansion of the corresponding rotation numbers in buric n mudrinic m and todorovic k 1997 30 l161 in the present paper this result is extended on the whole class of sufficiently smooth area preserving twist maps of cylinders we present numerical evidence that the and d q are the same for all critical invariant circles of any such map which have the rotation numbers with the same tail

time dependent cross correlations between different stock returns a directed network of influence we study the time dependent cross correlations of stock returns i e we measure the correlation as the function of the time shift between pairs of stock return time series using tick by tick data we find a weak but significant effect showing that in many cases the maximum correlation appears at nonzero time shift indicating directions of influence between the companies due to the weakness of this effect and the shortness of the characteristic time of the order of a few minutes our findings are compatible with market efficiency the interaction of companies defines a directed network of influence