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

analog study of the first passage time problem driven by power law distributed noise abstract in this work we develop an analog circuit to generate a stochastic signal with stationary distribution exhibiting a tunable power law tail the proposed circuit design is a variant of a recently introduced one based on a differential equation with both multiplicative and additive noises here the circuit is carefully designed in order to ensure that all components operate within their specific regimes we provide a detailed characterization of the output signal including the power law exponent dependence on the tunable component we apply the present circuit to study the first passage time problem in a simple integrate fire model of neural dynamics driven by a non gaussian noise

harmonic oscillator with multiplicative noise nonmonotonic dependence on the strength and the rate of dichotomous noise the output signal of a undamped linear oscillator with a random frequency subject to a periodic force shows nonmonotonic dependence on the strength and the rate of color noise stochastic resonance the effect is absent for white noise

a polynomial algorithm for the k cut problem for fixed k the k cut problem is to find a partition of an edge weighted graph into k nonempty components such that the total edge weight between components is minimum this problem is np complete for an arbitrary k and its version involving fixing a vertex in each component is np hard even for k 3 we present a polynomial algorithm for k fixed that runs in onki 2 3k 2 4tn m steps where tn m is the running time required to find the minimum s t cut on a graph with n vertices and m edges

a microscopic model of triangular arbitrage we introduce a microscopic model which describes the dynamics of each dealer in multiple foreign exchange markets taking account of the triangular arbitrage transaction the model reproduces the interaction among the markets well we explore the relation between the parameters of the present microscopic model and the spring constant of a macroscopic model that we proposed previously

a geometrical study of 3d incompressible euler flows with clebsch potentials a long lived euler flow and its power law energy spectrum abstract a simple initial condition for vorticity sin y z sin z x sin x y which has clebsch potentials has been identified to lead to a flow evolution with a very weak energy transfer this allows us to integrate the euler equations in time longer than commonly expected to reach a stage at which the total enstrophy attains its peak for the corresponding navier stokes flow it thereby enables us to study the relationship between the inviscid limit and totally inviscid behaviours numerically in spite of small energy dissipation rate the navier stokes flow shows a power law spectrum whose exponent is around 5 3 and 2 a similar behaviour is also observed for the euler flow in physical space this flow has groups of vorticity layers which hesitate to roll up

multiplicative stochastic differential equations with noise induced transitions the author investigates a class of linear multiplicative stochastic differential equations and demonstrates the existence of a striking noise induced transition in the structure of the resulting asymptotic stationary probability distribution for the dependent variable the transition amounts to a change from a bounded distribution to an unbounded one with only a finite number of convergent moments it occurs when the range of fluctuation of one of the variables driven by the underlying stochastic process increases sufficiently to permit changes of sign for the variable it seems likely that the phenomenon is a general one and occurs in a wider class of models than that discussed in this paper he obtains explicit results for simple cases which he confirms by appropriate numerical simulations this gives him the opportunity of assessing the applicability of perturbation theory which is one of the few calculational methods employed on these models up until now

statistical properties of deterministic threshold elements the case of market price abstract we analyze statistical properties of a set of deterministic threshold elements which is introduced as a model for the stock market the macroscopic variable of the stock market price shows seemingly stochastic fluctuation with a f 2 power spectrum consistent with real economic fluctuations the maximum lyapunov exponent is estimated to be zero indicating that the system is at the edge of chaos

an eulerian lagrangian approach for incompressible fluids local theory we study a formulation of the incompressible euler equations in terms of the inverse lagrangian map in this formulation the equations become a first order advective nonlinear system of partial differential equations

approximation and hardness results for the max k uncut problem abstract in the study of the homophily law of large scale complex networks we get a combinatorial optimization problem which we call the max k uncut problem given an n vertex undirected graph g v e with nonnegative weights w e e e defined on edges and a positive integer k the max k uncut problem asks to find a partition v 1 v 2 v k of v such that the total weight of edges that are not cut is maximized intuitively an edge that is not cut connects two vertices with the same or similar attributes since they are in the same part of the partition interestingly the max k uncut problem is just the complement of the classic min k cut problem for max k uncut we present a randomized 1 k n 2 approximation algorithm a greedy 1 2 k 1 n approximation algorithm and an 1 2 approximation algorithm by reducing it to densest k subgraph where is the approximation ratio of the densest k subgraph problem more importantly we show that max k uncut and densest k subgraph are in fact equivalent in approximability up to a factor of 2 we also prove an approximation hardness result for max k uncut under the assumption p np

dynamic numerical models of stock market price from microscopic determinism to macroscopic randomness abstract a variant of threshold dynamics is introduced to model the behaviors of a large assembly of dealers in a stock market although the microscopic evolution dynamics is deterministic the collective behaviors such as market prices show seemingly stochastic fluctuations the statistical properties of market price change can be well approximated by a simple discrete langevin type equation with random amplification the macroscopic stochastic equation is solved both numerically and analytically showing that the market price change generally follow power law distributions in the steady state the reason for the appearance of rapid decay in the distribution tails are discussed

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

probability distributions for one component equations with multiplicative noise abstract systems described by equations involving both multiplicative and additive noise are common in nature examples include convection of a passive scalar field polymers in turbulent flow and noise in dye lasers in this paper the one component version of this problem is studied the steady state probability distribution is classified into two different types of behavior one class has power law tails and the other is of the form of an exponential to a power law the value of the power law exponent is determined analytically for models having colored gaussian noise it is found to only depend on the power spectrum of the noise at zero frequency when non gaussian noise is considered it is shown that stretched exponential tails are possible an intuitive understanding of the results is found and makes use of the lyapunov exponents for these systems

frequency analysis of tick quotes on foreign currency markets and the double threshold agent model abstract power spectrum densities for the number of tick quotes per minute market activity on three currency markets usd jpy eur usd and jpy eur are analyzed for periods from january 2000 to december 2000 we find some peaks on the power spectrum densities at a few minutes we develop the double threshold agent model and confirm that the corresponding periodicity can be observed for the activity of this model even though market participants perceive common weaker periodic information than threshold for decision making of them this model is numerically performed and theoretically investigated by utilizing the mean field approximation we propose a hypothesis that the periodicities found on the power spectrum densities can be observed due to nonlinearity and diversity of market participants

finding k cuts within twice the optimal two simple approximation algorithms for the minimum k cut problem are presented each algorithm finds a k cut having weight within a factor of 2 2 k of the optimal one of our algorithms is particularly efficient it requires a total of only n 1 maximum flow computations for finding a set of near optimal k cuts one for each value of k between 2 and n

note on loss of regularity for solutions of the 3 d incompressible euler and related equations one of the central problems in the mathematical theory of turbulence is that of breakdown of smooth indefinitely differentiable solutions to the equations of motion in 1934 j leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3 d incompressible navier stokes equations the problem is still open we show in this report that breakdown of smooth solutions to the 3 d incompressible slightly viscous i e corresponding to high reynolds numbers or highly turbulent navier stokes equations cannot occur without breakdown in the corresponding solution of the incompressible euler ideal fluid equation we prove then that solutions of distorted euler equations which are equations closely related to the euler equations for short term intervals do breakdown

volatility cluster and herding stock markets can be characterized by fat tails in the volatility distribution clustering of volatilities and slow decay of their time correlations for an explanation models with several mechanisms and consequently many parameters as the lux marchesi model have been used we show that a simple herding model with only four parameters leads to a quantitative description of the data as a new type of data we describe the volatility cluster by the waiting time distribution which can be used successfully to distinguish between different models

kinetic theory for financial brownian motion from microscopic dynamics recent technological development has enabled researchers to study social phenomena scientifically in detail and financial markets has particularly attracted physicists since the brownian motion has played the key role as in physics in our previous report arxiv 1703 06739 to appear in phys rev lett we have presented a microscopic model of trend following high frequency traders hfts and its theoretical relation to the dynamics of financial brownian motion directly supported by a data analysis of tracking trajectories of individual hfts in a financial market here we show the mathematical foundation for the hft model paralleling to the traditional kinetic theory in statistical physics we first derive the time evolution equation for the phase space distribution for the hft model exactly which corresponds to the liouville equation in conventional analytical mechanics by a systematic reduction of the liouville equation for the hft model the bogoliubov born green kirkwood yvon hierarchal equations are derived for financial brownian motion we then derive the boltzmann like and langevin like equations for the order book and the price dynamics by making the assumption of molecular chaos the qualitative behavior of the model is asymptotically studied by solving the boltzmann like and langevin like equations for the large number of hfts which is numerically validated through the monte carlo simulation our kinetic description highlights the parallel mathematical structure between the financial brownian motion and the physical brownian motion

algorithmic aspects of homophyly of networks abstract we investigate the algorithmic problems of the homophyly phenomenon in networks given an undirected graph g v e and a vertex coloring c v 1 2 k of g we say that a vertex v v is happy if v shares the same color with all its neighbors and unhappy otherwise and that an edge e e is happy if its two endpoints have the same color and unhappy otherwise supposing c is a partial vertex coloring of g we define the maximum happy vertices problem mhv for short as to color all the remaining vertices such that the number of happy vertices is maximized and the maximum happy edges problem mhe for short as to color all the remaining vertices such that the number of happy edges is maximized let k be the number of colors allowed in the problems we show that both mhv and mhe can be solved in polynomial time if k 2 and that both mhv and mhe are np hard if k 3 we devise a max 1 k 3 approximation algorithm for the mhv problem where is the maximum degree of vertices in the input graph and a 1 2 approximation algorithm for the mhe problem this is the first theoretical progress of these two natural and fundamental new problems

statistical properties of the noisy on off intermittency the dependence of the escape time on the deviation from the critical point is investigated for the noisy on off intermittency the power law scaling behaviour is destroyed by the noise in a wide region above the critical point in the general case the width of the region decreases logarithmically with the decrease in the noise amplitude below the critical point the noise induced escape time is sensitive to the statistical properties of the chaotic driving signal universal exponential dependence holds only for the gaussian approximation

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

multifractal features of financial markets abstract we study the tick dynamical behavior of three assets in financial markets the kospi the won dollar and yen dollar exchange rates using the rescaled range r s analysis the multifractal hurst exponents with long run memory effect can be obtained from those assets and we discuss whether there exists the crossover or not for the hurst exponents at characteristic time scales particularly we find that the probability distribution of returns approaches to a lorentz distribution different from gaussian properties

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

free bits pcps and nonapproximability towards tight results this paper continues the investigation of the connection between probabilistically checkable proofs pcps and the approximability of np optimization problems the emphasis is on proving tight nonapproximability results via consideration of measures such as the free bit complexity and the amortized free bit complexity of proof systems the first part of the paper presents a collection of new proof systems based on a new error correcting code called the long code we provide a proof system that has amortized free bit complexity of 2 epsilon implying that approximating maxclique within n frac13 e and approximating the chromatic number within n frac15 e are hard assuming np neq corp for any e 0 we also derive the first explicit and reasonable constant hardness factors for min vertex cover msat 2 and max cut and we improve the hardness factor for msat 3 we note that our nonapproximability factors for maxsnp problems are appreciably close to the values known to be achievable by polynomial time algorithms finally we note a general approach to the derivation of strong nonapproximability results under which the problem reduces to the construction of certain gadgets the increasing strength of nonapproximability results obtained via the pcp connection motivates us to ask how far this can go and whether pcps are inherent in any way the second part of the paper addresses this the main result is a reversal of the connection due to feige et al fglss connection j acm 43 1996 pp 268 292 where the latter had shown how to translate proof systems for np into np hardness of approximation results for maxclique we show how any np hardness of approximation result for maxclique yields a proof system for np roughly our result says that for any constant f if maxclique is np hard to approximate within n 1 1 f then np subseteq overline fpcp log f the latter being the class of languages possessing proofs of logarithmic randomness and amortized free bit complexity f this suggests that pcps are inherent to obtaining nonapproximability results furthermore the tight relation suggests that reducing the amortized free bit complexity is necessary for improving the nonapproximability results for maxclique the third part of our paper initiates a systematic investigation of the properties of pcp and fpcp free pcp as a function of the following various parameters randomness query complexity free bit complexity amortized free bit complexity proof size etc we are particularly interested in triviality results which indicate which classes are not powerful enough to capture np we also distill the role of randomized reductions in this area and provide a variety of useful transformations between proof checking complexity classes

power law distributions in random multiplicative processes with non gaussian colored multipliers abstract one class of universal mechanisms that generate power law probability distributions is that of random multiplicative processes in this paper we consider a multiplicative langevin equation driven by non gaussian colored multipliers we analytically derive a formula that relates the power law exponent to the statistics of the multipliers and numerically confirm its validity using multiplicative noise generated by chaotic dynamical systems and by a two valued markov process we also investigate the relationship between our treatment and the large deviation analysis of time series and demonstrate the appearance of log periodic fluctuations superimposed on the power law distribution due to the non gaussian nature of the multipliers

on off intermittency power spectrum and fractal properties of time series abstract some dynamical systems possess invariant submanifolds such that the dynamics restricted to the invariant submanifold is chaotic this situation arises in systems with a spatial symmetry or in the synchronization of chaotic oscillators the invariant submanifold could become unstable to perturbations in the transverse directions when a parameter of the system is changed through a critical blow out value this could result in an extreme form of temporally intermittent bursting called on off intermittency we propose a model that incorporates the universal features of systems that display on off intermittency we study this model both with and without additive noise and we derive scaling results for the power spectral density of the on off intermittent process and for the box counting dimension for the set of time intervals when the process takes on values above a given threshold we then present numerical simulations realizing these results

the interacting gaps model reconciling theoretical and numerical approaches to limit order models we consider the emergence of power law tails in the returns distribution of limit order driven markets we explain a previously observed clash between the theoretical and numerical studies of such models we introduce a solvable model that interpolates between the previous studies and agrees with each of them in the relevant limit c