on the cauchy problem and initial traces for a degenerate parabolic equation the authors study the cauchy problem for the degenerate parabolic equation ut div du p 2 du p 2 and find sufficient conditions on the initial trace u0 and in particular on its behaviour as x for existence of a solution in some strip rn 0 t using a harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions uniqueness of solutions is shown if u0 belongs to l1loc rn but is left open in the case that u0 is merely a locally bounded measure the results are closely related to papers by aronson caffarelli benilan crandall pierre and dahlberg kenig about the porous medium equation ut um the proofs are different and allow one to generalize some of the above results to equations with variable coefficients

global existence and l estimates of solution for doubly nonlinear parabolic equation the existence uniqueness and l estimate of the weak solution to the initial boundary value problem for the doubly nonlinear parabolic equationut div u r u m 2 u au fx t uin 0 with zero boundary condition in a bounded domain rn are established in particular the behavior of the solution as t 0 and t is investigated

a note on blow up of solutions of a quasilinear heat equation with vanishing initial energy abstract in this work we consider an initial boundary value problem related to the equation u t div u m 2 u f u and prove under suitable conditions on f a blow up result for solutions with vanishing or negative initial energy

quantifying and interpreting collective behavior in financial markets firms having similar business activities are correlated we analyze two different cross correlation matrices c constructed from i 30 min price fluctuations of 1000 us stocks for the two year period 1994 95 and ii one day price fluctuations of 422 us stocks for the 35 year period 1962 96 we find that the eigenvectors of c corresponding to the largest eigenvalues allow us to partition the set of all stocks into distinct subsets these subsets are similar to business sectors and are stable for extended periods of time we find that price fluctuations of these subsets are characterized by power law decaying time correlations reminiscent of strongly interacting systems the internal structure of a complex system manifests itself in correlations among its constituents in complex physical systems interactions between constituents cause collective modes having special statistical properties which reflect the underlying dynamics can we quantify collective movement of stock prices in analogous terms to address this question we analyze the equal time correlation matrix c constructed from the price fluctuations of a large number of stocks first we find that the collective modes for the stock market problem partition the set of all stocks studied into distinct subsets typically these subsets are formed by combinations of related industries and in some cases they go beyond grouping by industry due to company diversification the traditional partitioning of firms into subsets by products and services is difficult and sometimes arbitrary and thus our results could be viewed as a statistical alternative to traditional industry classification 1 furthermore we find that the price fluctuations of the collective modes display long range power law time correla

an asymptotic and numerical description of self similar blow up in quasilinear parabolic equations we study the blow up behaviour of two reaction diffusion problems with a quasilinear degenerate diffusion and a superlinear reaction we show that in each case the blow up is self similar in contrast to the linear diffusion limit of each in which the diffusion is only approximately self similar we then investigate the limit of the self similar behaviour and describe the transition from a stable manifold blow up behaviour for quasilinear diffusion to a centre manifold one for the linear diffusion

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

on solutions to the initial boundary value problem for an equation in the theory of infiltration etude de l existence et du comportement des solutions du probleme de valeur limite initiale

smoothing effects of the initial boundary value problem for logarithmic type quasilinear parabolic equations abstract we give existence theorems of global solutions in l l o c 0 w 0 1 to the initial boundary value problem for quasilinear degenerate parabolic equations of the form u t div u 2 u 0 where the class of v 2 includes the logarithmic case u 2 log 1 u 2 for a typical example we assume that the initial data belong to w 0 1 p 0 p 0 2 or l r r 1 and we derive precise estimates for u t near t 0

a dynamical system approach to the self similar large time behavior in scalar convection diffusion equations abstract we present a dynamical system method that provides both the existence of self similar solutions and the self similar large time behavior for convection diffusion equations in r n this method avoids a direct study of the elliptic problem related to the self similar profiles we concentrate our attention on the model example u t u a u 1 n u g x t in r n 0 u 0 u 0 l 1 r n l r n where a r n and g x t t 1 n 2 1h x formula with h l2 r n exp x 2 4 and l r n such that h y dy 0 and h y y dy being small enough in the natural similarity variables the self similar profiles become stationary solutions of a new convection diffusion equation by using lyapunov type arguments that rely in an essential manner on the l1 contraction property of the system we prove that those stationary solutions exists and that any trajectory converges to one of them the limit of any trajectory is completely determined by its mass which is conserved along the time in order to ensure the relative compactness of trajectories we work in the functional framework of the weighted sobolev spaces introduced by escobedo and kavian

random matrix approach to cross correlations in financial data we analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory rmt using two large databases we calculate cross correlation matrices c of returns constructed from i 30 min returns of 1000 us stocks for the 2 yr period 1994 1995 ii 30 min returns of 881 us stocks for the 2 yr period 1996 1997 and iii 1 day returns of 422 us stocks for the 35 yr period 1962 1996 we test the statistics of the eigenvalues lambda i of c against a null hypothesis a random correlation matrix constructed from mutually uncorrelated time series we find that a majority of the eigenvalues of c fall within the rmt bounds lambda lambda for the eigenvalues of random correlation matrices we test the eigenvalues of c within the rmt bound for universal properties of random matrices and find good agreement with the results for the gaussian orthogonal ensemble of random matrices implying a large degree of randomness in the measured cross correlation coefficients further we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the rmt bound display systematic deviations from the rmt prediction in addition we find that these deviating eigenvectors are stable in time we analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks our analysis of the remaining deviating eigenvectors shows distinct groups whose identities correspond to conventionally identified business sectors finally we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return less

on global attractor for m laplacian parabolic equation with local and nonlocal nonlinearity abstract in this paper we study the long time behavior of solutions for m laplacian parabolic equation u t m u a x u u f 0 u k y u y t d y g x in 0 with the initial data u x 0 u 0 x l q q 1 and zero boundary condition in two cases for a x a 0 0 and a x 0 are considered we obtain the existence and l p estimate of global attractor a in l p for any p max 1 q the attractor a is in fact a bounded set in w 0 1 m l if a x a 0 0 in and a is bounded in w 0 1 m l p if a x 0 in

structure of a financial cross correlation matrix under attack abstract we investigate the structure of a perturbed stock market in terms of correlation matrices for the purpose of perturbing a stock market two distinct methods are used namely local and global perturbation the former involves replacing a correlation coefficient of the cross correlation matrix with one calculated from two gaussian distributed time series while the latter reconstructs the cross correlation matrix just after replacing the original return series with gaussian distributed time series concerning the local case it is a technical study only and there is no attempt to model reality the term global means the overall effect of the replacement on other untouched returns through statistical analyses such as random matrix theory rmt network theory and the correlation coefficient distributions we show that the global structure of a stock market is vulnerable to perturbation however apart from in the analysis of inverse participation ratios iprs the vulnerability becomes dull under a small scale perturbation this means that these analysis tools are inappropriate for monitoring the whole stock market due to the low sensitivity of a stock market to a small scale perturbation in contrast when going down to the structure of business sectors we confirm that correlation based business sectors are regrouped in terms of iprs this result gives a clue about monitoring the effect of hidden intentions which are revealed via portfolios taken mostly by large investors

pseudo paths towards minimum energy states in network dynamics abstract the dynamics of networks forming on heider balance theory moves towards lower tension states the condition derived from this theory enforces agents to reevaluate and modify their interactions to achieve equilibrium these possible changes in network s topology can be considered as various paths that guide systems to minimum energy states based on this theory the final destination of a system could reside on a local minimum energy jammed state or the global minimum energy balanced states the question we would like to address is whether jammed states just appear by chance or there exist some pseudo paths that bound a system towards a jammed state we introduce an indicator to suspect the location of a jammed state based on the inverse participation ratio method ipr we provide a margin before a local minimum where the number of possible paths dramatically drastically decreases this is a condition that proves adequate for ending up on a jammed states

global attractors and convergence to equilibrium for degenerate ginzburg landau and parabolic equations abstract we study the dynamics of a degenerate complex ginzburg landau cgl and a real parabolic equation with a variable generally nonsmooth diffusion coefficient which may vanish at some points or be unbounded for the cgl equation there exists a global attractor in l 2 for the parabolic equation we show the existence of a global branch of nonnegative stationary states the global bifurcation result is used in order to establish in conjunction with the definition of a gradient dynamical system in the natural phase space that any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium

gradient estimates for degenerate diffusion equations ii we establish upper and lower bounds for various norms of solutions and their gradients for the equation u t div u m 1 u in n in terms of the norms of the initial data based on the l estimate of u we conclude that u x t is lipschitz continuous in space time for all t 0 whenever u x 0 is in l 1 n

quantifying and modeling long range cross correlations in multiple time series with applications to world stock indices we propose a modified time lag random matrix theory in order to study time lag cross correlations in multiple time series we apply the method to 48 world indices one for each of 48 different countries we find long range power law cross correlations in the absolute values of returns that quantify risk and find that they decay much more slowly than cross correlations between the returns the magnitude of the cross correlations constitutes bad news for international investment managers who may believe that risk is reduced by diversifying across countries we find that when a market shock is transmitted around the world the risk decays very slowly we explain these time lag cross correlations by introducing a global factor model gfm in which all index returns fluctuate in response to a single global factor for each pair of individual time series of returns the cross correlations between returns or magnitudes can be modeled with the autocorrelations of the global factor returns or magnitudes we estimate the global factor using principal component analysis which minimizes the variance of the residuals after removing the global trend using random matrix theory a significant fraction of the world index cross correlations can be explained by the global factor which supports the utility of the gfm we demonstrate applications of the gfm in forecasting risks at the world level and infinding uncorrelated individual indices we find ten indices that are practically uncorrelated with the global factor and with the remainder of the world indices which is relevant information for world managers in reducing their portfolio risk finally we argue that this general method can be applied to a wide range of phenomena in which time series are measured ranging from seismology and physiology to atmospheric geophysics

global existence to the initial boundary value problem for a system of nonlinear diffusion and wave equations abstract we prove the global existence of weak solution pair to the initial boundary value problem for a system of m laplacian type diffusion equation and nonlinear wave equation the interaction of two equations is given through nonlinear source terms f u v and g u v

microstructural degeneracy associated with a two point correlation function and its information content a two point correlation function provides a crucial yet an incomplete characterization of a microstructure because distinctly different microstructures may have the same correlation function in an earlier letter gommes jiao and torquato phys rev lett 108 080601 2012 we addressed the microstructural degeneracy question what is the number of microstructures compatible with a specified correlation function we computed this degeneracy i e configurational entropy in the framework of reconstruction methods which enabled us to map the problem to the determination of ground state degeneracies here we provide a more comprehensive presentation of the methodology and analyses as well as additional results since the configuration space of a reconstruction problem is a hypercube on which a hamming distance is defined we can calculate analytically the energy profile of any reconstruction problem corresponding to the average energy of all microstructures at a given hamming distance from a ground state the steepness of the energy profile is a measure of the roughness of the energy landscape associated with the reconstruction problem which can be used as a proxy for the ground state degeneracy the relationship between this roughness metric and the ground state degeneracy is calibrated using a monte carlo algorithm for determining the ground state degeneracy of a variety of microstructures including realizations of hard disks and poisson point processes at various densities as well as those with known degeneracies e g single disks of various sizes and a particular crystalline microstructure we show that our results can be expressed in terms of the information content of the two point correlation functions from this perspective the a priori condition for a reconstruction to be accurate is that the information content expressed in bits should be comparable to the number of pixels in the unknown microstructure we provide a formula to calculate the information content of any two point correlation function which makes our results broadly applicable to any field in which correlation functions are employed

on the initial boundary value problem for some quasilinear parabolic equations of divergence form abstract in this paper we give an existence theorem of global classical solution to the initial boundary value problem for the quasilinear parabolic equations of divergence form u t div u 2 u f u u x t where u 2 may not be bounded as u as an application the logarithmic type nonlinearity u 2 log 1 u 2 which is growing as u and degenerate at u 0 is considered under f 0

global solutions for some nonlinear parabolic equations with nonmonotonic peturbations publisher summary this chapter discusses the existence uniqueness and decay property of the solutions of two problems it also presents the sobolev s lemma and gagliardo nirenberg inequality

a note on on the existence of solutions of quasilinear wave equations with viscosity abstract in the note we study the existence and uniqueness of the global generalized solution for the following initial boundary value problem 1 u t t t x div u t x 2 u t x u t x u t t x u t t x p 1 u t t x u t x q 1 u t x x t 0 t 2 u t x 0 t 0 t 3 u 0 x u 0 x u t 0 x u 1 x x where is a bounded domain in r n with smooth boundary p 1 q 1 0 0 denotes the laplacian operator in r n denotes the gradient operator and s is a given nonlinear function

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

gradient estimates for solutions of parabolic equations and systems 1 statement of the problem let 52 c r n 2 be a bounded domain with smooth boundary r 2 in this note we study scalar quasilinear parabolic differential equations u div y g ivul vu f u 0 1 u a u f u o 2 and systems u da u f u 0 3 on q x 0 co with initial conditions u 0 u0 resp u 0 u0 and homogeneous dirichlet boundary conditions u i x o co o for 1 and 2 resp 4 u1r x 0 00 0 for 3 5

cauchy problem of a nonlocal p laplacian evolution equation with nonlocal convection abstract this paper is concerned with the cauchy problem of a nonlocal equation that takes into account convective and p laplacian diffusive effects u t x t r n j x y u y t u x t p 2 u y t u x t d y g f u f u x t with j radially symmetric and g not necessarily symmetric first we prove the existence and uniqueness of solutions and if the convolution kernels j and g are rescaled appropriately we show that solutions of the nonlocal problem converge to the solution of the usual p laplacian diffusion equation with convection finally as a supplementary result we study the asymptotic behavior of solutions as t and give the decay estimate

on global attractor for nonlinear parabolic equations of m laplacian type abstract existence and some regularity results of global attractor in l q q 1 for m laplacian type quasilinear parabolic equation with a perturbation like a x u u u u f x with 0 a x 0 are proved for the proofs moser s technique is used extensively

removable singularities of solutions of degenerate nonlinear elliptic equations on the boundary of a domain the removability of singularities of solutions for the dirichlet problem for degenerate nonlinear elliptic equations on the boundary of a domain is studied a method based on a priori energetic estimates of solutions to elliptic boundary value problems is used the growth in the vicinity of a boundary point finite or at infinity for generalized solutions is studied

stochastic dynamical model for stock stock correlations we propose a model of coupled random walks for stock stock correlations the walks in the model are coupled via a mechanism that the displacement price change of each walk stock is activated by the price gradients over some underlying network we assume that the network has two underlying structures describing the correlations among the stocks of the whole market and among those within individual groups respectively each with a coupling parameter controlling the degree of correlation the model provides the interpretation of the features displayed in the distribution of the eigenvalues for the correlation matrix of real market on the level of time sequences we verify that such modeling indeed gives good fitting for the market data of us stocks

estimation of the influence of second and third order moments on random sets reconstructions abstract to characterize a random image we often limit ourselves to the use of two functions the first one is the histogram or grey level distribution the other is the covariance function which allows the study of the spatial grey level distribution one evaluation of the relevance of these two functions to describe the morphology of images is presented especially we show the importance of having recourse to the centered third order moment in addition to these two functions to obtain a fine characterization of images this study is done for binary random images generated from the same primary grains poisson polygons or discs that are implanted in different ways we simulate with the gagalowicz s procedure some textures respecting the same second or third order moments we study the resemblance in term of the morphology described with erosion and dilation curves and with granulometry by opening and pseudo granulometry by closing with squares the analysis of the results of these measurements is carried out with the help of correspondence analysis that allows a quantitative and qualitative approach of the differences of morphology

stabilization of solutions for a class of degenerate equations in divergence form in one space dimension abstract by extending the concept of the liapunov functional and an idea due to dafermos we establish an invariance principle for a class of not necessarily strongly monotone systems this principle is then applied to continuous contraction strongly monotone systems on a hubert space and to a class of l 1 contraction monotone semigroups generated by degenerate equations in divergence form in one space dimension

collective behavior of stock price movements a random matrix theory approach we review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory rmt developed in physics to describe the energy levels ofcomplex systems rmt makes predictions f universal properties that do not depend on the interactions between the elements comprising the system deviations from rmt provide clues regarding system speci c properties we compare the statistics ofthe cross correlation matrix c whose elements cij are the correlation coe cients ofprice uctuations ofstock i and j against a random matrix having the same symmetry properties it is found that rmt methods can distinguish random and non random parts of c the non random part of c which deviates from rmt results provides information regarding genuine collective behavior among stocks c 2001 published by elsevier science b v

reconstructing random media we formulate a procedure to reconstruct the structure of general random heterogeneous media from limited morphological information by extending the methodology of rintoul and torquato j colloid interface sci bold 186 467 1997 developed for dispersions the procedure has the advantages that it is simple to implement and generally applicable to multidimensional multiphase and anisotropic structures furthermore an extremely useful feature is that it can incorporate any type and number of correlation functions in order to provide as much morphological information as is necessary for accurate reconstruction we consider a variety of one and two dimensional reconstructions including periodic and random arrays of rods various distribution of disks debye random media and a fontainebleau sandstone sample we also use our algorithm to construct heterogeneous media from specified hypothetical correlation functions including an exponentially damped oscillating function as well as physically unrealizable ones copyright ital 1998 ital the american physical society

global existence and gradient estimates for the quasilinear parabolic equations of m laplacian type with a nonlinear convection term abstract in this paper we derive precise estimates for u t including smoothing effects near t 0 and decay as t as well as global existence of the solutions u t to the initial boundary value problem in a bounded domain in r n for the quasilinear parabolic equation of the m laplacian type with a nonlinear convection term b u u for the initial data u 0 we only assume u 0 l q 1 q

contractive properties for the heat equation in sobolev spaces abstract consider the heat equation r u x u 0 in a cylinder 0 t r n 1 smooth lateral boundary under zero neumann or dirichlet conditions geometric conditions for are given that guarantee that for a given p x u t l p will be non increasing for any solution decay rates are also given for arbitrary and p it is shown how to construct an equivalent l p norm such that x t is non increasing in this norm

spatial size distributions applications to shape and texture analysis this paper proposes new descriptors for binary and gray scale images based on newly defined spatial size distributions ssd the main idea consists of combining a granulometric analysis of the image with a comparison between the geometric covariograms for binary images or the auto correlation function for gray scale images of the original image and its granulometric transformation the usual granulometric size distribution then arises as a particular case of this formulation examples are given to show that in those cases in which a finer description of the image is required the more complex descriptors generated from the ssd could be advantageously used it is also shown that the new descriptors are probability distributions so their intuitive interpretation and properties can be appropriately studied from the probabilistic point of view the usefulness of these descriptors in shape analysis is illustrated by some synthetic examples and their use in texture analysis is studied various cases of ssd and several former methods for texture classification are compared

estimates on the support of the solutions of some nonlinear elliptic and parabolic problems we obtain existence and uniqueness of solutions with compact support for some nonlinear elliptic and parabolic problems including the equations of one dimensional motion of a non newtonian fluid precise estimates for the support of these solutions are obtained and the optimality of our hypotheses is discussed

comparing the structure of an emerging market with a mature one under global perturbation a b s t r a c t in this paper we investigate the tehran stock exchange tse and dow jones industrial average djia in terms of perturbed correlation matrices to perturb a stock market there are two methods namely local and global perturbation in the local method we replace a correlation coefficient of the cross correlation matrix with one calculated from two gaussian distributed time series whereas in the global method we reconstruct the correlation matrix after replacing the original return series with gaussian distributed time series the local perturbation is just a technical study we analyze these markets through two statistical approaches random matrix theory rmt and the correlation coefficient distribution by using rmt we find that the largest eigenvalue is an influence that is common to all stocks and this eigenvalue has a peak during financial shocks we find there are a few correlated stocks that make the essential robustness of the stock market but we see that by replacing these return time series with gaussian distributed time series the mean values of correlation coefficients the largest eigenvalues of the stock markets and the fraction of eigenvalues that deviate from the rmt prediction fall sharply in both markets by comparing these two markets we can see that the djia is more sensitive to global perturbations these findings are crucial for risk management and portfolio selection

network trending leadership followership and neutrality among companies a random matrix approach abstract in this article we analyze the cross correlation between returns of different stocks to answer the following important questions the first one is if there exists collective behavior in a financial market how could we detect it and the second question is is there a particular company among the companies of a market as the leader of the collective behavior or is there no specified leadership governing the system similar to some complex systems we use the method of random matrix theory to answer the mentioned questions cross correlation matrix of index returns of four different markets is analyzed the participation ratio quantity related to each matrices eigenvectors and the eigenvalue spectrum is calculated we introduce shuffled matrix created of cross correlation matrix in such a way that the elements of the later one are displaced randomly comparing the participation ratio quantities obtained from a correlation matrix of a market and its related shuffled one on the bulk distribution region of the eigenvalues we detect a meaningful deviation between the mentioned quantities indicating the collective behavior of the companies forming the market by calculating the relative deviation of participation ratios we obtain a measure to compare the markets according to their collective behavior answering the second question we show there are three groups of companies the first group having higher impact on the market trend called leaders the second group is followers and the third one is the companies who have not a considerable role in the trend the results can be utilized in portfolio construction