u sets as a possibilistic set theory abstract the motivation of possibility theory is to represent the imprecision intrinsic in natural language in particular in expressions of the form x is r a suitable topos of presheaves i e an extension of classical set theory is presented where sets of the form x x is r can be constructed with x is r having the meaning given in possibility theory the aim of the paper is to give an explicit mathematical description of the context sensitivity of linguistic imprecision this can be done by defining sets that vary over the possible contexts i e sets in a topos of presheaves defined over a category that represents the flow of information

fractal properties of financial markets abstract we present an analysis of the usa stock market using a simple fractal function financial bubbles preceding the 1987 2000 and 2007 crashes are investigated using the besicovitch ursell fractal function fits show a good agreement with the s p 500 data when a complete financial growth is considered starting at the threshold of the abrupt growth and ending at the peak moving the final time of the fitting interval towards earlier dates causes growing discrepancy between two curves on the basis of a detailed analysis of the financial index behavior we propose a method for identifying the stage of the current financial growth and estimating the time in which the index value is going to reach the maximum

eigenvalue localization and neville elimination abstract neville elimination is an elimination procedure alternative to gaussian elimination and very adequate when dealing with some special classes of matrices in this paper we present pivoting strategies such that the radii of the gersgorin circles of the schur complements through neville elimination with these pivoting strategies reduce their length and we consider classes of matrices important in many applications we include illustrative examples comparing the results with those obtained with gaussian elimination and showing that our hypotheses are necessary

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

stock market and motion of a variable mass spring we establish an analogy between the motion of spring whose mass increases linearly with time and volatile stock market dynamics within an economic model based on simple temporal demand and supply functions e canessa j phys a 33 2000 3637 the total system energy et is shown to be proportional to a decreasing time dependent spring constant kt this model allows to derive log periodicity cos log t tc on commodity prices and oscillations surplus and shortages in the level of stocks we also made an attempt to connect these results to the tsallis statistics parameter q based on a possible force entropy correlation e canessa physica a 341 2004 165 and find that the tsallis second entropic term i 1wpiq q 1 relates to the square of the demand or supply function

can log periodic power law structures arise from random fluctuations abstract recent research has established log periodic power law lppl patterns prior to the detonation of the german stock index dax bubble in 1998 the purpose of this article is to explore whether a langevin equation extracted from real world data can generate synthetic time series with comparable lppl structures to this end we first estimate the stochastic process underlying the dax log returns during the period from mid 1997 until end 2003 the employed data set contains about 3 93 10 6 intraday dax quotes at a sampling rate of 15 s our results indicate that the dax log returns can be described as a markov process as a consequence a langevin equation is derived based on this model equation we run extensive simulations in order to generate 100 synthetic dax trajectories each covering 3000 trading days we find lppl behavior in ten artificial time series moreover we can establish a link between lppl patterns and ensuing bubble bursts in seven synthetic 600 week windows however the lppl components in most synthetic trajectories differ fundamentally from those lppl structures that have previously been detected in real financial time series summarized this paper demonstrates that lppl structures are not necessarily the signature of imitative behavior among investors but can also stem from noise even though the likelihood of this is extremely low thus our findings confirm with high statistical confidence that the lppl structures in the dax development are rooted deeper than only in the random fluctuations of the german stock market

the dominant degree and disc theorem for the schur complement of matrix the theory of schur complement plays an important role in many fields such as control theory and computational mathematics in this paper we obtain some estimates for the diagonally c diagonally and product c diagonally dominant degree on the schur complement of matrices which improve some relative results as application we present some bounds for the eigenvalues of schur complement by the entries of the original matrix instead of those of the schur complement particularly we obtain that the eigenvalues of the schur complements are located in the gerschgorin circles of the original matrices under certain conditions

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

punctuated evolution due to delayed carrying capacity a new delay equation is introduced to describe the punctuated evolution of complex nonlinear systems a detailed analytical and numerical investigation provides the classification of all possible types of solutions for the dynamics of a population in the four main regimes dominated respectively by i gain and competition ii gain and cooperation iii loss and competition and iv loss and cooperation our delay equation may exhibit bistability in some parameter range as well as a rich set of regimes including monotonic decay to zero smooth exponential growth punctuated unlimited growth punctuated growth or alternation to a stationary level oscillatory approach to a stationary level sustainable oscillations finite time singularities as well as finite time death

visibility graph network analysis of gold price time series abstract mapping time series into a visibility graph network the characteristics of the gold price time series and return temporal series and the mechanism underlying the gold price fluctuation have been explored from the perspective of complex network theory the network degree distribution characters which change from power law to exponent law when the series was shuffled from original sequence and the average path length characters which change from l ln n into ln l ln n as the sequence was shuffled demonstrate that price series and return series are both long rang dependent fractal series the relations of hurst exponent to the power law exponent of degree distribution demonstrate that the logarithmic price series is a fractal brownian series and the logarithmic return series is a fractal gaussian series power law exponents of degree distribution in a time window changing with window moving demonstrates that a logarithmic gold price series is a multifractal series the power law average clustering coefficient demonstrates that the gold price visibility graph is a hierarchy network the hierarchy character in light of the correspondence of graph to price fluctuation means that gold price fluctuation is a hierarchy structure which appears to be in agreement with elliot s experiential wave theory on stock price fluctuation and the local rule growth theory of a hierarchy network means that the hierarchy structure of gold price fluctuation originates from persistent short term factors such as short term speculation

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

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

eigenvalue localization and pivoting strategies for gaussian elimination we show pivoting strategies such that the radii of the gersgorin circles of the schur complements through gaussian elimination with these pivoting strategies reduce their length the results will be illustrated with classes of matrices important in many applications

possibilistic reasoning a mini survey and uniform semantics abstract in this paper we survey some quantitative and qualitative approaches to uncertainty management based on possibility theory and present a logical framework to integrate them the semantics of the logic is based on the dempster s rule of conditioning for possibility theory it is then shown that classical modal logic conditional logic possibilistic logic quantitative modal logic and qualitative possibilistic logic are all sublogics of the presented logical framework in this way we can formalize and generalize some well known results about possibilistic reasoning in a uniform semantics moreover our uniform framework is applicable to nonmonotonic reasoning approximate consequence relation formulation and partial consistency handling

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

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

the schur complement of ph matrices abstract in this paper we consider the class of p h matrices a subclass of h matrices and using scaling characterization we show that this class is closed under taking the schur complement we show that under certain conditions the perron complement of a p h matrix is a p h matrix we also present a way of constructing a scaling matrix for the given p h matrix and we give eigenvalue localization for the schur complement of a p h matrix using only the entries of the original matrix we illustrate this by numerical examples

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

nonextensive statistical features of the polish stock market fluctuations the statistics of return distributions on various time scales constitutes one of the most informative characteristics of the financial dynamics here we present a systematic study of such characteristics for the polish stock market index wig20 over the period 04 01 1999 31 10 2005 for the time lags ranging from 1min1min up to 1 h this market is commonly classified as emerging still on the shortest time scales studied we find that the tails of the return distributions are consistent with the inverse cubic power law as identified previously for majority of the mature markets within the time scales studied a quick and considerable departure from this law towards a gaussian can however be traced interestingly all the forms of the distributions observed can be comprised by the single q gaussians which provide a satisfactory and at the same time compact representation of the distribution of return fluctuations over all magnitudes of their variation the corresponding nonextensivity parameter q was found to systematically decrease when increasing the time scales the temporal correlations quantified here in terms of multifractality provide further arguments in favor of nonextensivity

discrete scale invariance in cross correlations between time series abstract the de trended cross correlation analysis dcca is converted to a new form which turns out to be a periodic function modulated power law to evaluate discrete scale long range cross correlation between time series if the modulator is dominated with one frequency the derived form will degenerate to a log periodic power law we investigate a total of five important stock markets distributing in different continents calculations show that the cross correlations between different stock markets may hint at log periodic oscillations this finding may be helpful for us to evaluate financial state in a global way

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 theoretical investigation into quantitative modal logic quantitative modal logic qml is a multi modal formulation of possibilistic reasoning with the capacity of representing and reasoning about higher order uncertainty in this paper some results about qml and possibility theory are investigated first the concept of filtration in classical modal logic is introduced into qml and the fundamental filtration theorem is proved second as a corollary the finite model property of qml is provided finally the theoretical and practical implications of finite model property are discussed

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

time scales involved in emergent market coherence abstract in addressing the question of the time scales characteristic for the market formation we analyze high frequency tick by tick data from the nyse and from the german market by using returns on various time scales ranging from seconds or minutes up to 2 days we compare magnitude of the largest eigenvalue of the correlation matrix for the same set of securities but for different time scales for various sets of stocks of different capitalization and the average trading frequency we observe a significant elevation of the largest eigenvalue with increasing time scale our results from the correlation matrix study can be considered as a manifestation of the so called epps effect there is no unique explanation of this effect and it seems that many different factors play a role here one of such factors is randomness in transaction moments for different stocks another interesting conclusion to be drawn from our results is that in the contemporary markets the emergence of significant correlations occurs on time scales much smaller than in the more distant history

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

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

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

a method for detection of abrupt changes in the financial market combining wavelet decomposition and correlation graphs abstract the objective of this work is to propose a new methodology to detect the imminence of abrupt changes in the stock market by combining a numerical indicator based on the wavelet decomposition technique with a measure of the interdependency of the markets using graph theory while the indicator based on wavelet decomposition is based on a single time series an approach based on network representation can provide information on the interdependency of the various markets more specifically the stock market indices are associated with nodes of a network and the correlation between pairs of nodes with links results from the theory of graphs can then be used to indicate numerically the connectivity of this network experimentations with a variety of financial time series shows that the connectivity varies as trends of the financial time series varies combining the indicator based on the wavelet decomposition with the proposed measure of the connectivity of the network it was possible to refine the authors previous results in terms of detecting abrupt changes in the stock market in order to illustrate the methodology a case study involving twelve stock market indices was presented

scaling technique for partition nekrasov matrices it is well known that for a given h matrix a there exists a diagonal nonsingular matrix that scales a by multiplying it from the right to a strictly diagonally dominant sdd matrix there are subclasses of h matrices that can be fully characterised by the form of the corresponding diagonal scaling matrices however for some applications it is not necessary to have such full characterisation it is sufficient to find at least one scaling matrix that will do the job the aim of this paper is to present a way of constructing a diagonal scaling matrix for one special subclass of h matrices called partition nekrasov matrices as an application of this scaling approach we obtain eigenvalue localisation for the corresponding schur complement matrix using only the entries of the original matrix

modeling symbiosis by interactions through species carrying capacities abstract we introduce a mathematical model of symbiosis between different species by taking into account the influence of each species on the carrying capacities of the others the modeled entities can pertain to biological and ecological societies or to social economic and financial societies our model includes three basic types symbiosis with direct mutual interactions symbiosis with asymmetric interactions and symbiosis without direct interactions in all cases we provide a complete classification of all admissible dynamical regimes the proposed model of symbiosis turned out to be very rich as it exhibits four qualitatively different regimes convergence to stationary states unbounded exponential growth finite time singularity and finite time death or extinction of species

eigenvalue localization refinements for the schur complement abstract the theory of schur complement plays an important role in many fields as well as the theory of h matrices in this paper we obtain some bounds for the eigenvalues of schur complement by the entries of the original matrix instead of those of the schur complement for some special h matrices also we show how this result for a wider class can be applied for estimating bounds for the eigenvalues of schur complement of an sdd matrix

further results on h matrices and their schur complements abstract it is well known d carlson t markham schur complements of diagonally dominant matrices czech math j 29 104 1979 246 251 1 that the schur complement of a strictly diagonally dominant matrix is strictly diagonally dominant also if a matrix is an h matrix then its schur complement is an h matrix too j liu y huang some properties on schur complements of h matrices and diagonally dominant matrices linear algebra appl 389 2004 365 380 8 recent research showed that the same type of statement holds for some special subclasses of h matrices see for example liu et al j liu y huang f zhang the schur complements of generalized doubly diagonally dominant matrices linear algebra appl 378 2004 231 244 the aim of this paper is to show that the proof of these results can be significantly simplified by using scaling approach as in zhang et al f zhang et al the schur complement and its applications springer new york 2005 and to give another invariance result of this type

pivoting strategies leading to diagonal dominance by rows pivoting strategies for gaussian elimination leading to upper triangular matrices which are diagonally dominant by rows are studied forward error analysis of triangular systems whose coefficient matrices are diagonally dominant by rows is performed we also obtain small bounds of the backward errors for the pivoting strategies mentioned above our examples of matrices include h matrices and some generalizations of diagonally dominant matrices and scaled partial pivoting for the 1 norm is an example of these pivoting strategies in the case of an n times n m matrix a pivoting strategy of computational complexity o n 2 is proposed which satisfies all the results of the paper