feedback cyclization over certain principal ideal domains definition de classes de paires atteignables sur des domaines principaux ideaux certains pour lesquelles la cyclisation de la retroaction est possible ceci conduit a une condition necessaire pour la cyclisation de la boucle de retour sous forme de resolution de quelques equations algebriques

power law periodicity in the tangent bifurcations of the logistic map abstract numerical studies were carried out for the average of the logistic map on the tangent bifurcations from chaos into periodic windows a critical exponent of 1 2 is found on the average amplitude as one approaches the transition additionally the averages oscillate with a period that decreases with the same exponent this power law periodicity is related to the reinjection mechanism of the map the undulations appear at control parameter values much earlier than the values where the critical exponent of the bifurcation shows significant changes in the average amplitude

p exponent and p leaders part i negative pointwise regularity multifractal analysis aims to characterize signals functions images or fields via the fluctuations of their local regularity along time or space hence capturing crucial features of their temporal spatial dynamics multifractal analysis is becoming a standard tool in signal and image processing and is nowadays widely used in numerous applications of different natures its common formulation relies on the measure of local regularity via the h older exponent by nature restricted to positive values and thus to locally bounded functions or signals it is here proposed to base the quantification of local regularity on p exponents a novel local regularity measure potentially taking negative values first the theoretical properties of p exponents are studied in detail second wavelet based multiscale quantities the p leaders are constructed and shown to permit accurate practical estimation of p exponents exploiting the potential dependence with p it is also shown how the collection of p exponents enriches the classification of locally singular behaviors in functions signals or images the present contribution is complemented by a companion article developing the p leader based multifractal formalism associated to p exponents

currency target zone modeling an interplay between physics and economics we study the performance of the euro swiss franc exchange rate in the extraordinary period from september 6 2011 to january 15 2015 when the swiss national bank enforced a minimum exchange rate of 1 20 swiss francs per euro within the general framework built on geometric brownian motions and based on the analogy between brownian motion in finance and physics the first order effect of such a steric constraint would enter a priori in the form of a repulsive entropic force associated with the paths crossing the barrier that are forbidden nonparametric empirical estimates of drift and volatility show that the predicted first order analogy between economics and physics is incorrect the clue is to realize that the random walk nature of financial prices results from the continuous anticipation of traders about future opportunities whose aggregate actions translate into an approximate efficient market with almost no arbitrage opportunities with the swiss national bank s stated commitment to enforce the barrier traders anticipation of this action leads to a vanishing drift together with a volatility of the exchange rate that depends on the distance to the barrier this effect is described by krugman s model p r krugman target zones and exchange rate dynamics q j econ 106 669 1991 we present direct quantitative empirical evidence that krugman s theoretical model provides an accurate description of the euro swiss franc target zone motivated by the insights from the economic model we revise the initial economics physics analogy and show that within the context of hindered diffusion the two systems can be described with the same mathematics after all using a recently proposed extended analogy in terms of a colloidal brownian particle embedded in a fluid of molecules associated with the underlying order book we derive that close to the restricting boundary the dynamics of both systems is described by a stochastic differential equation with a very small constant drift and a linear diffusion coefficient as a side result we present a simplified derivation of the linear hydrodynamic diffusion coefficient of a brownian particle close to a wall

influence of strong noise on the decline and propagation of population in the delayed malthus verhulst model abstract the effects of strong noise on the decline and propagation processes of a population in the malthus verhulst model with time delay are investigated by a stochastic simulation time delays in two different processes are concurrent in ecosystems the simulation results indicate that the stability of the population is enhanced by the decreasing multiplicative noise intensity and the increasing delay time the replacement of old individuals with young ones is accelerated by an increasing multiplicative noise intensity an increasing additive noise intensity and a decreasing delay time an increasing multiplicative noise intensity will drive the population of species to fluctuate more largely

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

cross correlated bounded noises induced the population extinction and enhancement of stability in a population growth model abstract the bounded noises as a new typed non gaussian noise have important practical significance in this paper the steady state and transient dynamical properties of the population growth model subjected to the cross correlated sine wiener bounded noise are investigated based on the approximate fokker planck equation the analytical expressions of stationary probability distribution function and extinction time of the population growth model are obtained the results show that the bounded noise intensity and the cross correlated bounded noise intensity can cause the population extinction in particular the population density are sensitive to the environmental fluctuation multiplicative noise and the self correlated time on the other hand the effects of noise intensities on the extinction time are discussed the results indicate that the extinction time can be decreased with the intrinsic stochasticity additive noise and the self correlated time increasing for the positive correlated case the result is similar to the additive noise and the self correlated time however for the negative correlated case the extinction time can be increased with the intensity increasing in particular the environmental fluctuation multiplicative noise causes noise enhanced stability for the negative correlated case and the extinction time firstly decreases for the small environmental fluctuation and then increases for the large environmental fluctuation for the positive correlated case the numerical results are in basic agreement with the theoretical predictions

new method for generating general solution of abel differential equation abstract a new method for generating a general solution of the nonlinear first kind abel type differential equation from a particular one is presented

renormalisation groups with periodic and aperiodic orbits motivated by recent results for some frustrated hierarchical models the authors analyse the consequences of renormalisation group transformations with periodic and aperiodic orbits they find that the free energy has in general an infinite number of singularities this occurs already when the transformation has a stable periodic orbit they present an idealised example and a new class of hierarchical potts models to which the theory applies

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

necessary and sufficient conditions for finite time singularities in ordinary differential equations abstract this paper gives a theorem which provides necessary and sufficient conditions for the existence of a real finite value of the independent variable at which the general solution of a certain class of ordinary differential equations diverges to infinity this class is a large subset of the set of all autonomous non linear polynomial ordinary differential equation these conditions involve the asymptotic form of the local series representation for the general solutions around the singularities and can be checked algorithmically

dynamics of solitons in a damped sine hilbert equation a damped sine hilbert sh equation is proposed it can be linearized by a dependent variable transformation which enables one to solve an initial value problem of the equation the n soliton solution is obtained explicitly and its properties are investigated in comparison with those of the n soliton solution of the sh equation in particular the interaction of the two solitons is explored in detail with the aid of the pole representation it is found that the interaction process is classified into the two types according to the initial amplitudes and positions of both solitons in the general n soliton case the long time behavior of the solution is shown to be characterized by the positive n zeros of the hermite polynomial of degree 2n finally a linearized version of the damped sh equation is briefly discussed

multi scaling mix and non universality between population and facility density the distribution of facilities is closely related to our social economic activities recent studies have reported a scaling relation between population and facility density with the exponent depending on the type of facility in this paper we show that generally this exponent is not universal for a specific type of facility instead by using chinese data we find that it increases with per capita gross domestic product gdp thus our observed scaling law is actually a mixture of several multi scaling relations this result indicates that facilities may change their public or commercial attributes according to the outside environment we argue that this phenomenon results from an unbalanced regional economic level and suggest a modification of a previous model by introducing the consuming capacity the modified model reproduces most of our observed properties c 2012 elsevier b v all rights reserved

singularity clustering in the duffing oscillator an asymptotic and numerical study is made of the singularity structure in the complex t plane of the duffing oscillator the presence of logarithmic terms in the local psi series expansion of the form t4 ln t leads to a multisheeted singularity structure of great complexity this structure is built recursively from an elemental pattern which takes the form of four armed stars of singularities this construction is deduced analytically from the properties of the mapping z t4 ln t and is confirmed quite accurately numerically a systematic resummation of the psi series in terms of lame functions is developed this series exhibits the same analytic structure at all orders and provides a semi local analytical representation of the solution which is apparently valid even in the chaotic regime

a model for prejudiced learning in noisy environments based on the heuristics that maintaining presumptions can be beneficial in uncertain environments we propose a set of basic requirements for learning systems to incorporate the concept of prejudice the simplest memoryless model of a deterministic learning rule obeying the axioms is constructed and shown to be equivalent to the logistic map the system s performance is analysed in an environment in which it is subject to external randomness weighing learning defectiveness against stability gained the corresponding random dynamical system with inhomogeneous additive noise is studied and shown to exhibit the phenomena of noise induced stability and stochastic bifurcations the overall results allow for the interpretation that prejudice in uncertain environments can entail a considerable portion of stubbornness as a secondary phenomenon

a spatial weighted network model based on optimal expected traffic abstract we propose a spatial weighted network model based on the optimal expected traffic the expected traffic represents the prediction of the flow created by two vertices and is calculated by the improved gravity equation the model maximizes the total expected traffic of the network by changing two parameters which control the fitness and the geographical constraints the model can vary its topology and give rise to a variety of statistical properties observed in the real world network notably our study shows that a linear and a nonlinear strength degree correlation can emerge when considering and neglecting the transport effect respectively

feedback invariants for linear dynamical systems over a principal ideal domain abstract this paper studies the action of the feedback group f n m on m input n dimensional reachable linear dynamical systems over a principal ideal domain r for such a 2 dimensional system a complete set of invariants is given which characterizes the feedback class of in particular it is characterized in terms of these invariants when has the feedback cyclization property the particular cases r z and r r x are studied in some detail finally when n is arbitrary the feedback classification is given for the class of reachable systems f g such that g is a matrix with at least n 1 invariant factors equal to one

log periodic self similarity an emerging financial law abstract a hypothesis that the financial log periodicity cascading self similarly through various time scales carries signatures of a law is pursued it is shown that the most significant historical financial events can be classified amazingly well using a single and unique value of the preferred scaling factor 2 which indicates that its real value should be close to this number this applies even to a declining decelerating log periodic phase crucial in this connection is identification of a super bubble bubble on bubble phenomenon identifying a potential universal preferred scaling factor as undertaken here may significantly improve the predictive power of the corresponding methodology several more specific related results include evidence that i the real end of the high technology bubble on the stock market started with a decelerating log periodic draw down in the beginning of september 2000 ii a parallel 2000 2002 decline seen in the standard poor s 500 from the log periodic perspective is already of the same significance as the one of the early 1930s and of the late 1970s iii all this points to a much more serious global crash in around 2025 of course from a level much higher at least one order of magnitude than in 2000

escape to infinity under the action of a potential and a constant electromagnetic field escape to infinity is proved for a great variety of potentials including the potential created by an infinite number of sources relativistic escape is studied escape in the presence of a constant electromagnetic field and a potential is also considered

kinks of the sine hilbert equation and their dynamical motions an exact method for constructing exact solutions of the sine hilbert equation is developed it is shown that the sh equation can be transformed into a system of non linear ordinary differential equations through a dependent variable transformation furthermore this system of equations is transformed into a system of decoupled linear ordinary differential equations exact solutions of the sh equation are then constructed by means of a simple integration the dynamical properties of the solutions are also studied in detail the solution is interpreted as a superposition of n pulses which may be termed kinks where n is a positive integer asymptotically for large time it is composed of a pulse with a constant velocity and n 1 static pulses this is a novel characteristic of the solution unlike the well known n soliton solution which behaves like n moving pulses for large time

growing networks under geographical constraints inspired by the structure of technological weblike systems we discuss network evolution mechanisms which give rise to topological properties found in real spatial networks thus we suggest that the peculiar structure of transport and distribution networks is fundamentally determined by two factors these are the dependence of the spatial interaction range of vertices on the vertex attractiveness or importance within the network and on the inhomogeneous distribution of vertices in space we propose and analyze numerically a simple model based on these generating mechanisms which seems for instance to be able to reproduce known structural features of the internet

a characterization of gcu rings with the fc2 property abstract we study linear systems f g with coefficients in a commutative ring r with the gcu property whenever f g is controllable there exists a vector u with gu unimodular our main result is a gcu ring is an fc2 ring i e every two dimensional controllable system over r is feedback cyclizable if and only if r satisfies the following property for any finitely generated ideal i and f invertible modulo i there exist a unit u and an element h with f uh2 mod i

kolmogorov s contributions to the physical and geometrical understanding of small scale turbulence and recent developments this paper reviews how kolmogorov postulated for the first time the existence of a steady statistical state for small scale turbulence and its defining parameters of dissipation rate and kinematic viscosity thence he made quantitative predictions of the statistics by extending previous methods of dimensional scaling to multiscale random processes we present theoretical arguments and experimental evidence to indicate when the small scale motions might tend to a universal form paradoxically not necessarily in uniform flows when the large scales are gaussian and isotropic and discuss the implications for the kinematics and dynamics of the fact that there must be singularities in the velocity field associated with the inertial range spectrum these may be particular forms of eddy or eigenstructure such as spiral vortices which may not be unique to turbulent flows also they tend to lead to the notable spiral contours of scalars in turbulence whose self similar structure enables the boxcounting technique to be used to measure the capacity dk of the contours themselves or of their intersections with lines dk although the capacity a term invented by kolmogorov and studied thoroughly by kolmogorov tikhomirov is like the exponent 2p of a spectrum in being a measure of the distribution of length scales d being related to 2p in the limit of very high reynolds numbers the capacity is also different in that experimentally it can be evaluated at local regions within a flow and at lower values of the reynolds number thus kolmogorov tikhomirov provide the basis for a more widely applicable measure of the self similar structure of turbulence finally we also review how kolmogorov s concept of the universal spatial structure of the small scales together with appropriate additional physical hypotheses enables other aspects of turbulence to be understood at these scales in particular the general forms of the temporal statistics such as the highfrequency inertial range spectra in eulerian and lagrangian frames of reference and the perturbations to the small scales caused by non isotropic non gaussian and inhomogeneous large scale motions 1 kolmogorov s papers review and comments a introduction in this review we join with the other contributors to this special publication in celebrating some of kolmogorov s great contributions to fluid mechanics and mathematics and showing in some small way how his genius has inspired further

damped finite time singularity driven by noise we consider the combined influence of linear damping and noise on a dynamical finite time singularity model for a single degree of freedom we find that the noise effectively resolves the finite time singularity and replaces it by a first passage time distribution or absorbing state distribution with a peak at the singularity and a long time tail the damping introduces a characteristic cross over time in the early time regime the probability distribution and first passage time distribution show a power law behavior with scaling exponent depending on the ratio of the nonlinear coupling strength to the noise strength in the late time regime the behavior is controlled by the damping the study might be of relevance in the context of hydrodynamics on a nanometer scale in material physics and in biophysics

finite time singularity in the dynamics of the world population economic and financial indices contrary to common belief both the earth s human population and its economic output have grown faster than exponential i e in a super malthusian mode for most of the known history these growth rates are compatible with a spontaneous singularity occurring at the same critical time 2052 10 signaling an abrupt transition to a new regime the degree of abruptness can be infered from the fact that the maximum of the world population growth rate was reached in 1970 i e about 80 years before the predicted singular time corresponding to approximately 4 of the studied time interval over which the acceleration is documented this rounding off of the finite time singularity is probably due to a combination of well known finite size effects and friction and suggests that we have already entered the transition region to a new regime as theoretical support a multivariate analysis coupling population capital r d and technology shows that a dramatic acceleration in the population growth during most of the timespan can occur even though the isolated dynamics do not exhibit it possible scenarios for the cross over and the new regime are discussed

a numerical approach for solving generalized abel type nonlinear differential equations in this paper a numerical power series algorithm which is based on the improved taylor matrix method is introduced for the approximate solution of abel type differential equations and also riccati differential equations the technique is defined and illustrated with some numerical examples the obtained results reveal that the method is very effective simple and valid high accuracy the method can be easily extended to other nonlinear equations

oscillatory finite time singularities in finance population and rupture abstract we present a simple two dimensional dynamical system where two nonlinear terms exerting respectively positive feedback and reversal compete to create a singularity in finite time decorated by accelerating oscillations the power law singularity results from the increasing growth rate the oscillations result from the restoring mechanism as a function of the order of the nonlinearity of the growth rate and of the restoring term a rich variety of behavior is documented analytically and numerically the dynamical behavior is traced back fundamentally to the self similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin the interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log periodic oscillations with remarkable scaling properties three domains of applications are discussed 1 the stock market with a competition between nonlinear trend followers and nonlinear value investors 2 the world human population with a competition between a population dependent growth rate and a nonlinear dependence on a finite carrying capacity 3 the failure of a material subjected to a time varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

escape to infinity in a newtonian potential escape to infinity in the presence of a newtonian potential is examined in the classical and relativistic cases

deterministic versus stochastic aspects of superexponential population growth models abstract deterministic population growth models with power law rates can exhibit a large variety of growth behaviors ranging from algebraic exponential to hyperexponential finite time explosion in this setup selfsimilarity considerations play a key role together with two time substitutions two stochastic versions of such models are investigated showing a much richer variety of behaviors one is the lamperti construction of selfsimilar positive stochastic processes based on the exponentiation of spectrally positive processes followed by an appropriate time change the other one is based on stable continuous state branching processes given by another lamperti time substitution applied to stable spectrally positive processes

finite time singularity signature of hyperinflation abstract we present a novel analysis extending the recent work of mizuno et al physica a 308 2002 411 on the hyperinflations of germany 1920 1 1 1923 11 1 hungary 1945 4 30 1946 7 15 brazil 1969 1994 israel 1969 1985 nicaragua 1969 1991 peru 1969 1990 and bolivia 1969 1985 on the basis of a generalization of cagan s model of inflation based on the mechanism of inflationary expectation of positive feedbacks between realized growth rate and people s expected growth rate we find that hyperinflations can be characterized by a power law singularity culminating at a critical time t c mizuno et al s double exponential function can be seen as a discrete time step approximation of our more general non linear ode formulation of the price dynamics which exhibits a finite time singular behavior this extension of cagan s model which makes natural the appearance of a critical time t c has the advantage of providing a well defined end of the clearly unsustainable hyperinflation regime we find an excellent and reliable agreement between theory and data for germany hungary peru and bolivia for brazil israel and nicaragua the super exponential growth seems to be already contaminated significantly by the existence of a cross over to a stationary regime

stochastic effects in mean field population growth the quasi gaussian approximation to the case of a taylor s law distributed substrate abstract we analyze the effect of the stochastic growth rates which may originate from heterogeneity of resource distribution on the average unrestricted growth of an ensemble of non interacting sub populations of consumers with holling i and ii functional responses assuming normal gaussian and anomalous tweedie density distributions of resources we show that the variation in the resource availability between the sub population habitats leads to a non malthusian dynamics of the initial phase of the average biomass growth and that this phase of growth can be successfully approximated using quasi gaussian approach this conclusion is illustrated by numerical simulations

power law periodicity in the 2003 2004 crude oil price dynamics abstract in recent months the crude oil market has shown an abnormal behavior as compared with the previous 20 years the most relevant feature of it being the high level prices well beyond the most pessimistic forecasting of a couple of years ago the aim of this paper is to show that the crude oil price dynamics in the period from mid 2003 to late 2004 can be described by a piecewise linear increasing trend plus an oscillatory behavior with power law periodicity in about mid december 2003 a transition in the period of the price oscillations was detected before and after such a date the oscillation period decreased and increased with time respectively additionally after july 2004 the slope of the trend has suffered a 200 increment which may indicate that the market instability is becoming stronger

spatial scaling in fracture propagation in dilute systems abstract the geometry of fracture patterns in a dilute elastic network is explored using molecular dynamics simulation the network in two dimensions is subjected to a uniform strain which drives the fracture to develop by the growth and coalescence of the vacancy clusters in the network for strong dilution it has been shown earlier that there exists a characteristic time t c at which a dynamical transition occurs with a power law divergence with the exponent z of the average cluster size close to t c the growth of the clusters is scale invariant in time and satisfies a dynamical scaling law this paper shows that the cluster growth near t c also exhibits spatial scaling in addition to the temporal scaling as fracture develops with time the connectivity length xi of the clusters increases and diverges at t c as xi t c t with 0 83 0 06 as a result of the scale invariant growth the vacancy clusters attain a fractal structure at t c with an effective dimensionality d f 1 85 0 05 these values are independent within the limit of statistical error of the concentration provided it is sufficiently high with which the network is diluted to begin with moreover the values are very different from the corresponding values in qualitatively similar phenomena suggesting a different universality class of the problem the values of and d f supports the scaling relation z d f with the value of z obtained before

renormalization group analysis of the 2000 2002 anti bubble in the us s p500 index explanation of the hierarchy of five crashes and prediction abstract we propose a straightforward extension of our previously proposed log periodic power law model of the anti bubble regime of the usa stock market since the summer of 2000 in terms of the renormalization group framework to model critical points using a previous work by gluzman and sornette phys rev e 65 2003 036142 on the classification of the class of weierstrass like functions we show that the five crashes that occurred since august 2000 can be accurately modeled by this approach in a fully consistent way with no additional parameters our theory suggests an overall consistent organization of the investors forming a collective network which interact to form the pessimistic bearish anti bubble regime with intermittent acceleration of the positive feedbacks of pessimistic sentiment leading to these crashes we develop retrospective predictions that confirm the existence of significant arbitrage opportunities for a trader using our model finally we offer a prediction for the unknown future of the us s p500 index extending over 2003 and 2004 that refines the previous prediction of sornette and zhou quant finance 2 2002 468

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

pole assignment for systems over rings a very simple proof of the pole assignment theorem for systems over a principal ideal domain and other rings is given furthermore an algorithm is presented extensions are also indicated

pole assignability in polynomial rings power series rings and prufer domains the topics treated in this paper have their origins in the area of algebraic systems theory however the paper itself should be classified as pure commutative algebra and we shall present it as such in the body of the text still it is appropriate to give a brief paragraph of motivation if a physical system is governed by a pair f g of matrices then the stability of the system can be determined by examining the eigenvalues of the matrix f if the system is unstable a feedback matrix k can sometimes be employed in such a way that the eigenvalues of the matrix f gk measure the stability of the modified system f gk g in this manner an unstable system can be rendered stable the pole assignability problem over commutative rings is one method of attacking the problem of finding such matrices k the paper is divided into four sections section 1 is given over almost entirely to defining the properties in which we shall be interested it concludes with a theorem about residuating and lifting the properties section 2 is concerned with the preservation of the properties under polynomial ring and power series ring formation section 3 is concerned with feedback cyclization a strong form of pole assignability section 4 is concerned with pole assignability over priifer domains a complete elaboration of our results must await the introduction of the necessary terminology for now we mention the following results in somewhat vague language if r is a zero dimensional ring then the pole assignability problem is solvable in r x and in r xl from a ringtheoretic standpoint almost any class of commutative rings contains members for which the feedback cyclization problem is solvable on the other hand if r is a ring with 1 in its stable range then the feedback cyclization problem is solvable in r if and only if a certain nice matricial property

two dimensional dynamical system associated with abel s nonlinear differential equation a two dimensional dynamical system is proposed that is described by a pair of nonlinear ordinary differential equations odes with a complex parameter it reduces to abel s nonlinear ode of the first kind by an appropriate transformation using this fact the properties of solutions are investigated in detail with the aid of numerical computations it is found that various types of bifurcation phenomena occur depending on the values of the parameter in particular the solution is shown to blow up in finite time under certain conditions in order to visualize the behaviors of dynamical motions the trajectories of solutions are depicted in the plane finally a discussion is made on some generalizations of the proposed system