a characterization of ideal polyhedra in hyperbolic 3 space the goals of this paper are to provide a characterization of dihedral angles of convex ideal those with all vertices on the sphere at infinity polyhedra in h3 and also of those convex polyhedra with some vertices on the sphere at infinity and some in the finite part of h3 these characterizations are given in respectively theorems 0 1 and 10 5 the first theorem is proved in detail while the proof of the second which is similar is only outlined the results of this paper grow out of the general framework of the author s doctoral dissertation 13 as published in 20 a lot of the language and some of the auxilary results come from there as well so familiarity with the latter reference is very helpful the necessity of the conditions postulated in theorem 0 1 has been shown in 14 hence only the sufficiency direction will be shown in the current paper in 1832 jakob steiner asked for a combinatorial characterization of convex polyhedra inscribed in the sphere this was considered intractable it took almost a hundred years to find a single example of an uninscribable combinatorial type by steinitz 24 however theorem 0 1 gives such a characterization section 11 is dedicated to a very brief historical survey and some purely graph theoretic and computational geometric consequences of theorem 0 1 in order to state this theorem suppose that a convex ideal polyhedron p in h3 is given let p denote the poincar dual of p and assign to each edge e of p a weight w e equal to the exterior dihedral angle at the corresponding edge e of p then the following result holds

singularity spectrum of self organized criticality i introduce a simple continuous probability theory based on the ginzburg landau equation that provides for the first time a common analytical basis to relate and describe the main features of two seemingly different phenomena of condensed matter physics namely self organized criticality and multifractality numerical support is given by a comparison with reported simulation data within the theory the origin of self organized critical phenomena is analysed in terms of a nonlinear singularity spectrum different from the typical convex shape due to multifractal measures

dynamical model and nonextensive statistical mechanics of a market index on large time windows the shape and tails of partial distribution functions pdf for a financial signal i e the s and the turbulent nature of the markets are linked through a model encompassing tsallis nonextensive statistics and leading to evolution equations of the langevin and fokker planck type a model originally proposed to describe the intermittent behavior of turbulent flows describes the behavior of normalized log returns for such a financial market index for small and large time windows and both for small and large log returns these turbulent market volatility of normalized log returns distributions can be sufficiently well fitted with a ensuremath chi 2 distribution the transition between the small time scale model of nonextensive intermittent process and the large scale gaussian extensive homogeneous fluctuation picture is found to be at ca a 200 day time lag the intermittency exponent ensuremath kappa in the framework of the kolmogorov log normal model is found to be related to the scaling exponent of the pdf moments thereby giving weight to the model the large value of ensuremath kappa points to a large number of cascades in the turbulent process the first kramers moyal coefficient in the fokker planck equation is almost equal to zero indicating no restoring force a comparison is made between normalized log returns and mere price increments

volatility clustering and scaling for financial time series due to attractor bubbling a microscopic model of financial markets is considered consisting of many interacting agents spins with global coupling and discrete time heat bath dynamics similar to random ising systems the interactions between agents change randomly in time in the thermodynamic limit the obtained time series of price returns show chaotic bursts resulting from the emergence of attractor bubbling or on off intermittency resembling the empirical financial time series with volatility clustering for a proper choice of the model parameters the probability distributions of returns exhibit power law tails with scaling exponents close to the empirical ones

multifractal behavior of the korean stock market index kospi abstract we investigate multifractality in the korean stock market index kospi the generalized qth order height height correlation function shows multiscaling properties there are two scaling regimes with a crossover time around t c 40 min we consider the original data sets and the modified data sets obtained by removing the daily jumps which occur due to the difference between the closing index and the opening index to clarify the origin of the multifractality we also smooth the data through convolution with a gaussian function after convolution we observe that the multifractality disappears in the short time scaling regime t t c but remains in the long time scaling regime t t c regardless of whether or not the daily jumps are removed we suggest that multifractality in the short time scaling regime is caused by the local fluctuations of the stock index but the multifractality in the long time scaling regime appears to be due to the intrinsic trading properties such as herding behavior information outside the market the long memory of the volatility and the nonlinear dynamics of the stock market

emerging properties of financial time series in the game of life we explore the spatial complexity of conway s game of life a prototypical cellular automaton by means of a geometrical procedure generating a two dimensional random walk from a bidimensional lattice with periodical boundaries the one dimensional projection of this process is analyzed and it turns out that some of its statistical properties resemble the so called stylized facts observed in financial time series the scope and meaning of this result are discussed from the viewpoint of complex systems in particular we stress how the supposed peculiarities of financial time series are often overrated in their importance

nonlinear complexity of random visibility graph and lempel ziv on multitype range intensity interacting financial dynamics abstract in an attempt to investigate the nonlinear complex evolution of financial dynamics a new financial price model the multitype range intensity contact mric financial model is developed based on the multitype range intensity interacting contact system in which the interaction and transmission of different types of investment attitudes in a stock market are simulated by viruses spreading two new random visibility graph vg based analyses and lempel ziv complexity lzc are applied to study the complex behaviors of return time series and the corresponding random sorted series the vg method is the complex network theory and the lzc is a non parametric measure of complexity reflecting the rate of new pattern generation of a series in this work the real stock market indices are considered to be comparatively studied with the simulation data of the proposed model further the numerical empirical study shows the similar complexity behaviors between the model and the real markets the research confirms that the financial model is reasonable to some extent

laminations mesurees de plissage des varietes hyperboliques de dimension 3 for a hyperbolic metric on a 3 dimensional manifold the boundary of its convex core is a surface which is almost everywhere totally geodesic but which is bent along a family of disjoint geodesics the locus and intensity of this bending is described by a measured geodesic lamination which is a topological object we consider two problems the topological characterization of those measured geodesic laminations which can occur as bending measured laminations of hyperbolic metrics and the uniqueness problem which asks whether a hyperbolic metric is uniquely determined by its bending measured lamination

financial multifractality and its subtleties an example of dax abstract detailed study of multifractal characteristics of the financial time series of asset values and of its returns is performed using a collection of the high frequency deutsche aktienindex data the tail index the renyi exponents based on the box counting algorithm for the graph d q and the generalized hurst exponents h q are computed in parallel for short and daily return times the results indicate a more complicated nature of the stock market dynamics than just consistent multifractal

stock market dynamics and turbulence parallel analysis of fluctuation phenomena abstract we report analogies and differences between the fluctuations in an economic index and the fluctuations in velocity of a fluid in a fully turbulent state specifically we systematically compare i the statistical properties of the s p 500 cash index recorded during the period january 84 december 89 with ii the statistical properties of the velocity of turbulent air measured in the atmospheric surface layer about 6 m above a wheat canopy in the connecticut agricultural research station we find non gaussian statistics and intermittency for both processes i and ii but the deviation from a gaussian probability density function are different for stock market dynamics and turbulence

multifractality in time series we apply the concepts of multifractal physics to financial time series in order to characterize the onset of crash for the standard poor 500 s p500 stock index x t it is found that within the framework of multifractality the analogous specific heat of the s p500 discrete price index displays a shoulder to the right of the main peak for low time lag values for decreasing t the presence of the shoulder is a consequence of the peaked temporal x t t x t fluctuations in this regime for large time lags t 80 we have found that cq displays typical features of a classical phase transition at a critical point an example of such dynamic phase transition in a simple economic model system based on a mapping with multifractality phenomena in random multiplicative processes is also presented by applying former results obtained with a continuous probability theory for describing scaling measures

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

statistical properties of the volatility of price fluctuations we study the statistical properties of volatility measured by locally averaging over a time window t the absolute value of price changes over a short time interval ensuremath delta t we analyze the s 500 stock index for the 13 year period jan 1984 to dec 1996 we find that the cumulative distribution of the volatility is consistent with a power law asymptotic behavior characterized by an exponent ensuremath mu ensuremath approx 3 similar to what is found for the distribution of price changes the volatility distribution retains the same functional form for a range of values of t further we study the volatility correlations by using the power spectrum analysis both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents also both methods show the presence of a crossover at approximately 1 5 days in addition we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 u s companies over the two year period jan 1994 to dec 1995

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

self organized criticality and stock market dynamics an empirical study abstract the stock market is a complex self interacting system characterized by intermittent behaviour periods of high activity alternate with periods of relative calm in the present work we investigate empirically the possibility that the market is in a self organized critical state soc a wavelet transform method is used in order to separate high activity periods related to the avalanches found in sandpile models from quiescent a statistical analysis of the filtered data shows a power law behaviour in the avalanche size duration and laminar times the memory process implied by the power law distribution of the laminar times is not consistent with classical conservative models for self organized criticality we argue that a near soc state or a time dependence in the driver which may be chaotic can explain this behaviour

a characterization of compact convex polyhedra in hyperbolic 3 space in this paper we study the extrinsic geometry of convex polyhedral surfaces in three dimensional hyperbolic spaceh 3 we obtain a number of new uniqueness results and also obtain a characterization of the shapes of convex polyhedra inh 3 in terms of a generalized gauss map this characterization greatly generalizes andre ev s theorem

tsallis statistics and gradually truncated levy flight distribution of an economical index abstract tsallis postulated a generalized form for entropy and give rise to a new statistics now known as tsallis statistics in the present work we compare the tsallis statistics with the gradually truncated levy flight and discuss the distribution of an economical index the standard and poor s 500 using the values of standard deviation as calculated by our model we find that both statistics give almost the same distribution thus we feel that gradual truncation of levy distribution after certain critical step size for describing complex systems is a requirement of generalized thermodynamics or similar the gradually truncated levy flight is based on physical considerations and bring a better physical picture of the dynamics of the whole system tsallis statistics gives a theoretical support both statistics together can be utilized for the development of a more exact portfolio theory or to understand better the complexities in human and financial behaviors a comparison of both statistics is made

volatility distribution in the s p500 stock index abstract we study the volatility of the s p500 stock index from 1984 to 1996 and find that the volatility distribution can be very well described by a log normal function further using detrended fluctuation analysis we show that the volatility is power law correlated with hurst exponent 0 9

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

scaling and correlation in financial time series we discuss the results of three recent phenomenological studies focussed on understanding the distinctive statistical properties of nancial time series i the probability distribution of stock price uctuations stock price uctuations occur in all magnitudes in analogy to earthquakes from tiny uctuations to very drastic events such as the crash of 19 october 1987 sometimes referred to as black monday the distribution of price uctuations decays with a power law tail well outside the levy stable regime and describes uctuations that dier by as much as 8 orders of magnitude in addition this distribution preserves its functional form for uctuations on time scales that dier by 3 orders of magnitude from 1 min up to approximately 10 days ii correlations in nancial time series while price uctuations themselves have rapidly decaying correlations the magnitude of uctuations measured by either the absolute value or the square of the price uctuations has correlations that decay as a power law persisting for several months iii volatility and trading activity we quantify the relation between trading activity measured by the number of transactions nt and the price change gt for a given stock over a time interval t t t we nd that nt displays long range power law correlations in time which leads to the interpretation that the long range correlations previously found for jgtj are connected to those of nt c 2000 elsevier science b v all rights reserved

algebraic limits of kleinian groups which rearrange the pages of a book in this paper we give examples of two new phenomena in kleinian groups we rst exhibit a sequence of homeomorphic marked hyperbolic 3 manifolds whose algebraic limit is not homeomorphic to any element in the sequence we then use this construction to exhibit situations where the space of convex co compact representations of a given 3 manifold group has many components but its closure is connected let m be a compact irreducible oriented 3 manifold and let d 1 m denote the space of all discrete faithful representations of 1 m into psl2 c a sequence of representations n d 1 m converging to d 1 m gives rise to a sequence n n h n 1 m of hyperbolic 3 manifolds each of which is homotopy equivalent to m the sequence n n is said to converge algebraically to n h 1 m see 7 13 14 for more information about algebraic convergence of kleinian groups in many situations see 1 6 15 24 25 27 it has been shown that n n must be homeomorphic to n for all large enough n and we had suspected that this would always be the case in this paper we give a collection of examples where n n is not homeomorphic to n for any n our sequences are quite well behaved the n 1 m are convex co compact and mutually quasiconformally conjugate and the algebraic limit 1 m is geometrically nite in our examples m is obtained by gluing a collection of i bundles to a solid torus along a family of parallel annuli these manifolds are particularly simple examples of books of i bundles see 9 where to explain the terminology one should think of the solid torus as the binding and the i bundles

stochastic cellular automata model for stock market dynamics in the present work we introduce a stochastic cellular automata model in order to simulate the dynamics of the stock market a direct percolation method is used to create a hierarchy of clusters of active traders on a two dimensional grid active traders are characterized by the decision to buy ensuremath sigma i t 1 or sell ensuremath sigma i t ensuremath 1 a stock at a certain discrete time step the remaining cells are inactive ensuremath sigma i t 0 the trading dynamics is then determined by the stochastic interaction between traders belonging to the same cluster extreme intermittent events such as crashes or bubbles are triggered by a phase transition in the state of the bigger clusters present on the grid where almost all the active traders come to share the same spin orientation most of the stylized aspects of the financial market time series including multifractal proprieties are reproduced by the model a direct comparison is made with the daily closures of the s index

crossover in the cont bouchaud percolation model for market fluctuations abstract monte carlo simulations of the cont bouchaud herding model for stock market traders show power law distributions for short times and exponential truncation for longer time intervals if they are made at the percolation threshold in two to seven dimensions

convergence of freely decomposable kleinian groups we consider a compact orientable hyperbolic 3 manifold with a compressible boundary suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination we prove that if this limit projective lamination is doubly incompressible then the sequence has compact closure in the deformation space as a consequence we generalise thurston s double limit theorem and solve his conjecture on convergence of function groups affirmatively

multiscale analysis from turbulent time series with wavelet transform abstract we present a multiscale signal analysis based on the multifractal spectrum obtained by the wavelet transform modulus maxima technique we analyze time series from turbulent data the first step is to obtain the pdf of the flutuations for velocities records and then to fit them by means of the tsallis generalized thermodynamics tsallis j stat phys 52 1998 479 the second step is to obtain the multifractal spectra of the time series by the wavelet transform muzy et al phys rew lett 67 1991 3515 the aim of this approach was to investigate a possible phenomenological connection between the entropic parameter q and the multifractal spectrum for turbulence

average bending of convex pleated planes in hyperbolic three space if p is a pleated plane in 3 dimensional hyperbolic space h 3 and a geodesic in its intrinsic metric we define b p the average bending of p in the direction we show that if p is a convex pleated plane embedded in h 3 then b p k for some universal k furthermore if p is a boundary component of the convex hull of a quasi fuchsian group then b p b almost everywhere where b is a constant times the length of the bending lamination of the pleated surface x p we use these to prove a number of results about quasi fuchsian groups including a universal bound on the lipschitz constant for the map to infinity and a bound on the length of by a constant times the euler characteristic of x

scaling of the distribution of fluctuations of financial market indices up to more than one month we find that the distributions for dt 4d 1560 min are consistent with a power law asymptotic behavior characterized by an exponent a 3 well outside the stable levy regime 0 a 2 to test the robustness of the s p result we perform a parallel analysis on two other financial market indices database iv contains 3560 daily records of the nikkei index for the 14 year period 1984 1997 and database v contains 4649 daily records of the hang seng index for the 18 year period 1980 1997 we find estimates ofa consistent with those describing the distribution of s p 500 daily returns one possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility for time scales longer than dt 3 4 d our results are consistent with a slow convergence to gaussian behavior s1063 651x 99 11211 x pacs number s 05 40 fb 05 45 tp 89 90 1n

nonextensive statistical mechanics and economics abstract ergodicity this is to say dynamics whose time averages coincide with ensemble averages naturally leads to boltzmann gibbs bg statistical mechanics hence to standard thermodynamics this formalism has been at the basis of an enormous success in describing among others the particular stationary state corresponding to thermal equilibrium there are however vast classes of complex systems which accommodate quite badly or even not at all within the bg formalism such dynamical systems exhibit in one way or another nonergodic aspects in order to be able to theoretically study at least some of these systems a formalism was proposed 14 years ago which is sometimes referred to as nonextensive statistical mechanics we briefly introduce this formalism its foundations and applications furthermore we provide some bridging to important economical phenomena such as option pricing return and volume distributions observed in the financial markets and the fascinating and ubiquitous concept of risk aversion one may summarize the whole approach by saying that bg statistical mechanics is based on the entropy s bg k i p i ln p i and typically provides exponential laws for describing stationary states and basic time dependent phenomena while nonextensive statistical mechanics is instead based on the entropic form s q k 1 i p i q q 1 with s 1 s bg and typically provides for the same type of description asymptotic power laws

scaling behaviors in differently developed markets scaling properties of four di erent stock market indices are studied in terms of a generalized hurst exponent approach we nd that the deviations from pure brownian motion behavior are associated with the degrees of development of the markets and we observe strong di erentiations in the scaling properties of markets at di erent development stage c

herd behavior in weight driven information spreading models for financial market abstract we study two weight driven information spreading models for financial market in these models we find that the activity threshold below which the financial crash occurs can be increased by uneven distribution of information weight compared with eguiluz and zimmermann model v m eguiluz m g zimmermann phys rev lett 85 2000 5659 we also find that below the threshold the normalized return distribution p z t satisfies p z 0 t exp t b whereas p z 0 t t above the threshold here t is the time interval where the normalized return is defined z t t z t t z t by approximating the relative increase of p z t 1 for large z as gaussian distribution with non zero mean we show that the non zero mean of the gaussian distribution can cause such exponentially decaying behavior of p z 0 t

dynamical generalized hurst exponent as a tool to monitor unstable periods in financial time series we investigate the use of the hurst exponent dynamically computed over a weighted moving time window to evaluate the level of stability instability of financial firms financial firms bailed out as a consequence of the 2007 2008 credit crisis show a neat increase with time of the generalized hurst exponent in the period preceding the unfolding of the crisis conversely firms belonging to other market sectors which suffered the least throughout the crisis show opposite behaviors we find that the multifractality of the bailed out firms increase at the crisis suggesting that the multi fractal properties of the time series are changing these findings suggest the possibility of using the scaling behavior as a tool to track the level of stability of a firm in this paper we introduce a method to compute the generalized hurst exponent which assigns larger weights to more recent events with respect to older ones in this way large fluctuations in the remote past are less likely to influence the recent past we also investigate the scaling associated with the tails of the log returns distributions and compare this scaling with the scaling associated with the hurst exponent observing that the processes underlying the price dynamics of these firms are truly multi scaling

line transversals of convex polyhedra in mathbb r 3 we establish a bound of o n 2k 1 varepsilon for any varepsilon 0 on the combinatorial complexity of the set mathcal t of line transversals of a collection mathcal p of k convex polyhedra in mathbb r 3 with a total of n facets and we present a randomized algorithm which computes the boundary of mathcal t in comparable expected time thus when k ll n the new bounds on the complexity and construction cost of mathcal t improve upon the previously best known bounds which are nearly cubic in n to obtain the above result we study the set mathcal t ell 0 of line transversals which emanate from a fixed line ell 0 establish an almost tight bound of o nk 1 varepsilon on the complexity of mathcal t ell 0 and provide a randomized algorithm which computes mathcal t ell 0 in comparable expected time slightly improved combinatorial bounds for the complexity of mathcal t ell 0 and comparable improvements in the cost of constructing this set are established for two special cases both assuming that the polyhedra of mathcal p are pairwise disjoint the case where ell 0 is disjoint from the polyhedra of mathcal p and the case where the polyhedra of mathcal p are unbounded in a direction parallel to ell 0 our result is related to the problem of bounding the number of geometric permutations of a collection mathcal c of k pairwise disjoint convex sets in mathbb r 3 namely the number of distinct orders in which the line transversals of mathcal c visit its members we obtain a new partial result on this problem