global well posedness of incompressible inhomogeneous fluid systems with bounded density or non lipschitz velocity in this paper we first prove the global existence of weak solutions to the d dimensional incompressible inhomogeneous navier stokes equations with initial data a 0 in l infty mathbb r d u 0 u 0 h u 0 d in dot b 1 frac d p p r mathbb r d which satisfy mu a 0 l infty u 0 h dot b 1 frac d p p r rm exp c r mu 2r u 0 d dot b 1 frac d p p r 2r leqq c 0 mu for some positive constants c 0 c r and 1 p d 1 r the regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non lipschitz velocity fields furthermore with additional regularity assumptions on the initial velocity or on the initial density we can also prove the uniqueness of such a solution we should mention that the classical maximal l p l q regularity theorem for the heat kernel plays an essential role in this context

maximal spanning trees asset graphs and random matrix denoising in the analysis of dynamics of financial networks abstract we study the time dependence of maximal spanning trees and asset graphs based on correlation matrices of stock returns in these networks the nodes represent companies and links are related to the correlation coefficients between them special emphasis is given to the comparison between ordinary and denoised correlation matrices the analysis of single and multi step survival ratios of the corresponding networks reveals that the ordinary correlation matrices are more stable in time than the denoised ones our study also shows that some information about the cluster structure of the companies is lost in the denoising procedure cluster structure that makes sense from an economic point of view exists and can easily be observed in networks based on denoised correlation matrices however this structure is somewhat clearer in the networks based on ordinary correlation matrices some technical aspects such as the random matrix denoising procedure are also presented

strong solutions to the cauchy problem of two dimensional incompressible fluid models of korteweg type abstract this paper studies the local existence of strong solutions to the cauchy problem of the incompressible fluid models of korteweg type with vacuum as far field density the corresponding 3d problem has been solved by tan and wang 2010 21 notice that the technique used by tan and wang fails treating the 2d case because the l p norm p 2 of the velocity u cannot be controlled in terms only of u and u here in the present paper we will use the framework of weighted approximation estimates introduced by liang 2015 14 for navier stokes equations to obtain the local existence of strong solutions provided the initial density does not decay very slowly at infinity with the compact support case included

on the decay and stability to global solutions of the 3 d inhomogeneous navier stokes equations in this chapter we consider the large time decay and stability to any given global smooth solution of the three dimensional inhomogeneous incompressible navier stokes equations

correlated wishart ensembles and chaotic time series we study the correlated wishart ensembles in the context of time series analysis we are interested in the statistics of eigenlevels viz variables associated with independent eigenmodes in the system the motivation of this work is to study the effect of time series correlations on the wishart ensembles in this connection we derive the level density and the two point function for the correlated wishart ensembles by using the binary correlation method using our analytic results we analyze spectra of autocovariance matrices derived from single variable stationary time series we consider the stochastic time series of gaussian variables with exponentially decaying correlations and time series of chaotic maps viz the arnold map the standard map and the stadium billiard map in both cases correlated time series are encountered and analyzed under the framework of random matrix theory it is shown that the eigenlevel statistics for the chaotic maps follow closely those of correlated wishart ensembles it is indicated that the presence of collective modes in the spectra of autocovariance matrices is related to the integrability of the system

information transfer through disordered media by diffuse waves we consider the information content h of a scalar multiple scattered diffuse wave field psi r and the information capacity c of a communication channel that employs diffuse waves to transfer the information through a disordered medium both h and c are shown to be directly related to the mesoscopic correlations between the values of psi r at different positions r in space arising due to the coherent nature of the wave for the particular case of a communication channel between two identical linear arrays of n 1 equally spaced transmitters or receivers receiver spacing a we show that the average capacity proportional variant n and obtain explicit analytic expressions for n in the limit of n infinity and kl infinity where k 2pi lambda lambda is the wavelength and l is the mean free path modification of the above results in the case of finite but large n and kl is discussed as well if the signal to noise ratio s n exceeds some critical value s n c n is a nonmonotonic function of a exhibiting maxima at ka mpi m 1 2 em leader for smaller s n ka mpi correspond to local minima while the absolute maximum of n is reached at some ka approximately s n 1 2 max as maximized over the receiver spacing a and the optimal normalized receiver spacing ka opt as the spacing maximizing both max n and ka opt scale as s n 1 2 for s n max n approximately log s n for s n s n c

local existence and uniqueness of strong solutions to the navier stokes equations with nonnegative density abstract in this paper we consider the initial boundary value problem to the nonhomogeneous incompressible navier stokes equations local strong solutions are established for any initial data 0 u 0 w 1 l h 0 1 with 1 and if 2 then the strong solution is unique the initial density is allowed to be nonnegative and in particular the initial vacuum is allowed the assumption on the initial data is weaker than the previous widely used one that 0 u 0 h 1 l h 0 1 h 2 and no compatibility condition is required

a lagrangian approach for the incompressible navier stokes equations with variable density we investigate the cauchy problem for the inhomogeneous navier stokes equations in the whole n dimensional space under some smallness assumption on the data we show the existence of global in time unique solutions in a critical functional framework the initial density is required to belong to the multiplier space of input amssym dot b n p 1 p 1 bbb r n in particular piecewise constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities using lagrangian coordinates is the key to our results as it enables us to solve the system by means of the basic contraction mapping theorem as a consequence conditions for uniqueness are the same as for existence 2012 wiley periodicals inc

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

multifractality in the random parameter model for multivariate time series abstract the random parameter model was proposed to explain the structure of the covariance matrix in problems where most but not all of the eigenvalues of the covariance matrix can be explained by random matrix theory in this article we explore the scaling properties of the model as observed in the multifractal structure of the simulated time series we use the wavelet transform modulus maxima technique to obtain the multifractal spectrum dependence with the parameters of the model the model shows a scaling structure compatible with the stylized facts for a reasonable choice of the parameter values

a new method to estimate the noise in financial correlation matrices companies belonging to the same industrial branch are subject to similar economical influences hence the time series of their stocks can show similar trends implying a correlation financial correlation matrices measure the unsystematic correlations between time series of stocks such information is important for risk management it has been found by laloux et al that the correlation matrices are noise dressed a major reason being the finiteness of the time series we present a new and alternative method to estimate this noise we introduce a power mapping of the elements in the correlation matrix which suppresses the noise and thereby effectively prolongs the time series neither further data processing nor additional input is needed to develop and test our method we use a model suggested by noh which can be viewed as a special case of a factor model in economics we perform numerical simulations for the time series and obtain correlation matrices we support the numerics by a qualitative analytical discussion with our approach different correlation structures buried under this noise can be detected our method is general and can be applied to all systems in which time series are measured

statistical properties of random density matrices statistical properties of ensembles of random density matrices are investigated we compute traces and von neumann entropies averaged over ensembles of random density matrices distributed according to the bures measure the eigenvalues of the random density matrices are analysed we derive the eigenvalue distribution for the bures ensemble which is shown to be broader then the quarter circle distribution characteristic of the hilbert schmidt ensemble for measures induced by partial tracing over the environment we compute exactly the two point eigenvalue correlation function

distributions of singular values for some random matrices the singular value decomposition is a matrix decomposition technique widely used in the analysis of multivariate data such as complex space time images obtained in both physical and biological systems in this paper we examine the distribution of singular values of low rank matrices corrupted by additive noise past studies have been limited to uniform uncorrelated noise using diagrammatic and saddle point integration techniques we extend these results to heterogeneous and correlated noise sources we also provide perturbative estimates of error bars on the reconstructed low rank matrix obtained by truncating a singular value decomposition 1999 the american physical society

the bulk of the stock market correlation matrix is not pure noise abstract we analyse the structure of the distribution of eigenvalues of the stock market correlation matrix with increasing length of the time series representing the price changes we use 100 highly capitalized stocks from the american market and relate the result to the corresponding ensemble of wishart random matrices it turns out that systematically more eigenvalues remain beyond the borders prescribed by this variant of the random matrix theory rmt this may indicate that even the bulk of the spectrum of the stock market correlation matrix carries some sort of correlations that are masked by a measurement noise when the time series used to construct the matrix are short we also study some other characteristics of the noisy eigensignals like their return distributions temporal correlations or their multifractal spectra and the results support the above conclusions

global strong solutions for the incompressible nematic liquid crystal flows with density dependent viscosity coefficient abstract this paper is concerned with an initial boundary value problem of the incompressible nematic liquid crystal flows with density dependent viscosity in a smooth bounded domain r 3 the global well posedness of strong solutions with large oscillations is established in vacuum provided u 0 l 2 d 0 l 2 is suitably small with arbitrary large initial density which extended the local strong solution by gao et al 12 to be a global one

global well posedness of the navier stokes equations of an inhomogeneous fluid in the half space with inflow boundary condition abstract in this paper we study the global solvability of the incompressible navier stokes equations of an inhomogeneous fluid in the half space with inflow boundary condition we prove the existence of a solution in l q l p framework when the initial density is close to a positive constant and the initial and boundary data of velocity are in some critical function spaces uniqueness holds true if the given data have slightly more regularity

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

random but not so much a parameterization for the returns and correlation matrix of financial time series abstract a parameterization that is a modified version of a previous work is proposed for the returns and correlation matrix of financial time series and its properties are studied this parameterization allows easy introduction of non stationarity and it shows several of the characteristics of the true observed realizations such as fat tails volatility clustering and a spectrum of eigenvalues of the correlation matrix that can be understood as an extension of random matrix theory results the predicted behavior of this parameterization for the eigenvalues is compared with the eigenvalues of brazilian assets and it is shown that those predictions fit the data better than random matrix theory

construction of point vortex equilibria via brownian ratchets a theory capable of producing equilibrium configurations of point vortices in the plane along with a numerical scheme to compute them is described the theory is formulated as a problem in linear algebra where one must find solutions to the matrix equation agz 0 where a is the 1 2 n nk1 n non normal configuration matrix obtained by requiring that all intervortical distances remain fixed and g2r n are the n vortex strengths for existence of an equilibrium a must have a non trivial nullspace we consider the singular values of a when this has one or more zero singular values the nullspace of a is non empty and an equilibrium exists for some choice of g new equilibrium configurations are found numerically by randomly depositing n points in the plane which generically gives rise to a configuration matrix a with empty nullspace using the sum of squares of the k smallest singular values of a as a ratchet we thermally fluctuate the configuration allowing each point to execute a random walk in the plane retaining only those configurations which reduce this quantity at the next step the configuration is thus driven to one with nullspace a zko0 these converged states are not necessarily nearby their initial configurations typically they are asymmetric and often we can drive the same initial state to several different equilibria a reverse ratchet method is also described which can produce initial conditions that would evolve to a specified equilibrium state once a converged final state is achieved the full singular value decomposition of a is used to calculate an optimal basis set for the nullspace of a and thus all allowable g the distribution of the singular values gives important information on the size of each equilibrium state as measured by frobenius norm their distance from each other spacing and density and how far a randomly chosen system of n points in the plane is from the nearest equilibrium configuration with a specified rank as well as its shannon entropy

data driven estimates of the number of clusters in multivariate time series an important problem in unsupervised data clustering is how to determine the number of clusters here we investigate how this can be achieved in an automated way by using interrelation matrices of multivariate time series two nonparametric and purely data driven algorithms are expounded and compared the first exploits the eigenvalue spectra of surrogate data while the second employs the eigenvector components of the interrelation matrix compared to the first algorithm the second approach is computationally faster and not limited to linear interrelation measures

one factor model for the cross correlation matrix in the vietnamese stock market abstract random matrix theory rmt has been applied to the analysis of the cross correlation matrix of a financial time series the most important findings of previous studies using this method are that the eigenvalue spectrum largely follows that of random matrices but the largest eigenvalue is at least one order of magnitude higher than the maximum eigenvalue predicted by rmt in this work we investigate the cross correlation matrix in the vietnamese stock market using rmt and find similar results to those of studies realized in developed markets us europe japan 9 18 as well as in other emerging markets 20 21 19 22 importantly we found that the largest eigenvalue could be approximated by the product of the average cross correlation coefficient and the number of stocks studied we demonstrate this dependence using a simple one factor model the model could be extended to describe other characteristics of the realistic data

measures of globalization based on cross correlations of world financial indices abstract the cross correlation matrix of daily returns of stock market indices in a diverse set of 37 countries worldwide was analyzed comparison of the spectrum of this matrix with predictions of random matrix theory provides an empirical evidence of strong interactions between individual economies as manifested by three largest eigenvalues and the corresponding set of stable non random eigenvectors the observed correlation structure is robust with respect to changes in the time horizon of returns ranging from 1 to 10 trading days and to replacing individual returns with just their signs this last observation confirms that it is correlations between signs and not absolute values of fluctuations which are mostly responsible for the observed effect negative changes in the index are somewhat more correlated than the positive ones also in our data set the reaction of asian stock indices to changes in european and american ones persists for about 3 days

decomposing the stock market intraday dynamics abstract the correlation matrix formalism is used to study temporal aspects of the stock market evolution this formalism allows to decompose the financial dynamics into noise as well as into some coherent repeatable intraday structures the present study is based on the high frequency deutsche aktienindex dax data over the time period between november 1997 and september 1999 and makes use of both the corresponding returns as well as volatility variations one principal conclusion is that a bulk of the stock market dynamics is governed by the uncorrelated noise like processes there exists however a small number of components of coherent short term repeatable structures in fluctuations that may generate some memory effects seen in the standard autocorrelation function analysis laws that govern fluctuations associated with those various components are different which indicates an extremely complex character of the financial fluctuations

effect of changing data size on eigenvalues in the korean and japanese stock markets abstract in this study we attempted to determine how eigenvalues change according to random matrix theory rmt in stock market data as the number of stocks comprising the correlation matrix changes specifically we tested for changes in the eigenvalue properties as a function of the number and type of stocks in the correlation matrix we determined that the value of the eigenvalue increases in proportion with the number of stocks furthermore we noted that the largest eigenvalue maintains its identical properties regardless of the number and type whereas other eigenvalues evidence different features

random matrix theory analysis of cross correlations in the us stock market evidence from pearson s correlation coefficient and detrended cross correlation coefficient abstract in this study we first build two empirical cross correlation matrices in the us stock market by two different methods namely the pearson s correlation coefficient and the detrended cross correlation coefficient dcca coefficient then combining the two matrices with the method of random matrix theory rmt we mainly investigate the statistical properties of cross correlations in the us stock market we choose the daily closing prices of 462 constituent stocks of s p 500 index as the research objects and select the sample data from january 3 2005 to august 31 2012 in the empirical analysis we examine the statistical properties of cross correlation coefficients the distribution of eigenvalues the distribution of eigenvector components and the inverse participation ratio from the two methods we find some new results of the cross correlations in the us stock market in our study which are different from the conclusions reached by previous studies the empirical cross correlation matrices constructed by the dcca coefficient show several interesting properties at different time scales in the us stock market which are useful to the risk management and optimal portfolio selection especially to the diversity of the asset portfolio it will be an interesting and meaningful work to find the theoretical eigenvalue distribution of a completely random matrix r for the dcca coefficient because it does not obey the marcenko pastur distribution

global solutions to the 3d incompressible nematic liquid crystal system abstract in this paper we consider the global well posedness of the cauchy problem of the 3d incompressible nematic liquid crystal system with initial data in the critical besov space b 2 1 1 2 r 3 b 2 1 3 2 r 3 in particular we prove that there exists a positive constant c 0 such that the nematic liquid crystal system has a unique global solution with initial data u 0 d 0 u 0 h u 0 3 d 0 which satisfies 1 1 d 0 d 0 b 2 1 3 2 1 u 0 h b 2 1 1 2 exp c 0 2 u 0 3 b 2 1 1 2 1 2 c 0 for some c 0 sufficiently small where d 0 is a constant vector with d 0 1 here and are two positive viscosity constants

emerging spectra of singular correlation matrices under small power map deformations correlation matrices are a standard tool in the analysis of the time evolution of complex systems in general and financial markets in particular yet most analysis assume stationarity of the underlying time series this tends to be an assumption of varying and often dubious validity the validity of the assumption improves as shorter time series are used if many time series are used this implies an analysis of highly singular correlation matrices we attack this problem by using the so called power map which was introduced to reduce noise its nonlinearity breaks the degeneracy of the zero eigenvalues and we analyze the sensitivity of the so emerging spectra to correlations this sensitivity will be demonstrated for uncorrelated and correlated wishart ensembles

economic fluctuations and statistical physics quantifying extremely rare and less rare events in finance one challenge of economics is that the systems treated by these sciences have no perfect metronome in time and no perfect spatial architecture crystalline or otherwise nonetheless as if by magic out of nothing but randomness one finds remarkably fine tuned processes in time we present an overview of recent research joining practitioners of economic theory and statistical physics to try to better understand puzzles regarding economic fluctuations one of these puzzles is how to describe outliers phenomena that lie outside of patterns of statistical regularity we review evidence consistent with the possibility that such outliers may not exist this possibility is supported by recent analysis of databases containing information about each trade of every stock r 2007 elsevier b v all rights reserved

on the global wellposedness to the 3 d incompressible anisotropic navier stokes equations corresponding to the wellposedness result 2 for the classical 3 d navier stokes equations ns with initial data in the scaling invariant besov space mathcal b 1 frac3p p infty here we consider a similar problem for the 3 d anisotropic navier stokes equations ans where the vertical viscosity is zero in order to do so we first introduce the besov sobolev type spaces mathcal b frac12 frac12 4 and mathcal b frac12 frac12 4 t then with initial data in the scaling invariant space mathcal b frac12 frac12 4 we prove the global wellposedness for ans provided the norm of initial data is small enough compared to the horizontal viscosity in particular this result implies the global wellposedness of ans with high oscillatory initial data 1 2

denoising of diffusion mri using random matrix theory we introduce and evaluate a post processing technique for fast denoising of diffusion weighted mr images by exploiting the intrinsic redundancy in diffusion mri using universal properties of the eigenspectrum of random covariance matrices we remove noise only principal components thereby enabling signal to noise ratio enhancements this yields parameter maps of improved quality for visual quantitative and statistical interpretation by studying statistics of residuals we demonstrate that the technique suppresses local signal fluctuations that solely originate from thermal noise rather than from other sources such as anatomical detail furthermore we achieve improved precision in the estimation of diffusion parameters and fiber orientations in the human brain without compromising the accuracy and spatial resolution

topological properties of stock networks based on minimal spanning tree and random matrix theory in financial time series abstract we investigated the topological properties of stock networks constructed by a minimal spanning tree we compared the original stock network with the estimated network the original network is obtained by the actual stock returns while the estimated network is the correlation matrix created by random matrix theory we found that the consistency between the two networks increases as more eigenvalues are considered in addition we suggested that the largest eigenvalue has a significant influence on the formation of stock networks

global regularity of 2d density patches for inhomogeneous navier stokes this paper is about lions open problem on density patches lions in mathematical topics in fluid mechanics vol 1 volume 3 of oxford lecture series in mathematics and its applications clarendon press oxford university press new york 1996 whether or not inhomogeneous incompressible navier stokes equations preserve the initial regularity of the free boundary given by density patches using classical sobolev spaces for the velocity we first establish the propagation of c 1 gamma regularity with 0 gamma 1 in the case of positive density furthermore we go beyond this to show the persistence of a geometrical quantity such as the curvature in addition we obtain a proof for c 2 gamma regularity

accuracy of pseudo inverse covariance learning a random matrix theory analysis for many learning problems estimates of the inverse population covariance are required and often obtained by inverting the sample covariance matrix increasingly for modern scientific data sets the number of sample points is less than the number of features and so the sample covariance is not invertible in such circumstances the moore penrose pseudo inverse sample covariance matrix constructed from the eigenvectors corresponding to nonzero sample covariance eigenvalues is often used as an approximation to the inverse population covariance matrix the reconstruction error of the pseudo inverse sample covariance matrix in estimating the true inverse covariance can be quantified via the frobenius norm of the difference between the two the reconstruction error is dominated by the smallest nonzero sample covariance eigenvalues and diverges as the sample size becomes comparable to the number of features for high dimensional data we use random matrix theory techniques and results to study the reconstruction error for a wide class of population covariance matrices we also show how bagging and random subspace methods can result in a reduction in the reconstruction error and can be combined to improve the accuracy of classifiers that utilize the pseudo inverse sample covariance matrix we test our analysis on both simulated and benchmark data sets

quantifying the dynamics of financial correlations abstract a novel application of the correlation matrix formalism to study dynamics of the financial evolution is presented this formalism allows to quantify the memory effects as well as some potential repeatable intraday structures in the financial time series the present study is based on the high frequency deutsche aktienindex dax data over the time period between november 1997 and december 1999 and demonstrates a power of the method in this way two significant new aspects of the dax evolution are identified i the memory effects turn out to be sizably shorter than what the standard autocorrelation function analysis seems to indicate and ii there exist short term repeatable structures in fluctuations that are governed by a distinct dynamics the former of these results may provide an argument in favour of the market efficiency while the latter one may indicate origin of the difficulty in reaching a gaussian limit expected from the central limit theorem in the distribution of returns on longer time horizons

global solution to the 3d inhomogeneous nematic liquid crystal flows with variable density abstract in this paper we investigate the global existence and uniqueness of solution to the 3d inhomogeneous incompressible nematic liquid crystal flows with variable density in the framework of besov spaces it is proved that there exists a global and unique solution to the nematic liquid crystal flows if the initial data 0 1 u 0 n 0 e 3 m b p 1 3 p 1 r 3 b p 1 3 p 1 r 3 b p 1 3 p r 3 with 1 p 6 and satisfies 0 1 m b p 1 3 p 1 u 0 b p 1 3 p 1 n 0 e 3 b p 1 3 p c for some small c 0 depending only on p here m b p 1 3 p 1 r 3 is the multiplier space of besov space b p 1 3 p r 3 using the lagrangian approach in danchin and mucha 2012 2013 7 8 is the key to our results

stability to the global large solutions of 3 d navier stokes equations in this chapter we consider the stability of global large solutions to the 3 d classical isotropic navier stokes equations in the anisotropic sobolev spaces

on the analysis of cross correlations in south african market data we report briefly on an application of random matrix theory to the analysis of sa financial market data an analysis of cross correlations in south african financial market data e print cond mat 0402389 correlation matrices c are constructed from 10 years of daily data for stocks listed on the johannesburg stock exchange from january 1993 to december 2002 spectral properties of c are tested against random matrix predictions we highlight some quantitative differences which arise when treating prices as existing only when measured as opposed to interpolating missing or illiquid trading days with a zero order hold

incompressible inhomogeneous fluids in bounded domains of r3 with bounded density abstract in this paper we study the incompressible inhomogeneous navier stokes equations in bounded domains of r 3 involving bounded density functions 1 a based on the corresponding theory of besov spaces on domains we first obtain the global existence of weak solutions u with initial data a 0 l u 0 b q s 1 3 q for 1 q 3 1 s furthermore with additional regularity assumptions on the initial velocity we also prove the uniqueness of such a solution it is a generalization of a result established by huang et al 2013 20 for the whole space r 3

fujita kato theorem for the 3 d inhomogeneous navier stokes equations abstract in this paper we prove the global existence and uniqueness of the solution with discontinuous density for the 3 d inhomogeneous navier stokes equations if the initial data 0 u 0 l r 3 h s r 3 with s 1 2 satisfies 0 c 0 0 c 0 u 0 h 1 2 e for some small e 0 depending only on c 0 c 0 furthermore we introduce the dual method to show that if u 0 l p r 3 for p 6 5 2 the velocity satisfies the decay estimate k u t l 2 c 1 t k 2 p for t 1 k 0 1 with p 3 2 1 p 1 2

between homogeneous and inhomogeneous navier stokes systems the issue of stability abstract we construct large velocity vector solutions to the three dimensional inhomogeneous navier stokes system the result is proved via the stability of two dimensional solutions with constant density under the assumption that initial density is point wisely close to a constant key elements of our approach are estimates in the maximal regularity regime and the lagrangian coordinates considerations are done in the whole r 3

on the wellposedness of three dimensional inhomogeneous navier stokes equations in the critical spaces we prove the local wellposedness of three dimensional incompressible inhomogeneous navier stokes equations with initial data in the critical besov spaces without assumptions of small density variation furthermore if the initial velocity field is small enough in the critical besov space dot b 1 2 2 1 mathbb r 3 this system has a unique global solution

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

the incompressible navier stokes equations in vacuum we are concerned with the existence and uniqueness issue for the inhomogeneous incompressible navier stokes equations supplemented with h 1 initial velocity and only bounded nonnegative density in contrast with all the previous works on that topics we do not require regularity or positive lower bound for the initial density or compatibility conditions for the initial velocity and still obtain unique solutions those solutions are global in the two dimensional case for general data and in the three dimensional case if the velocity satisfies a suitable scaling invariant smallness condition as a straightforward application we provide a complete answer to lions question in 25 page 34 concerning the evolution of a drop of incompressible viscous fluid in the vacuum

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

random matrix approach to categorical data analysis correlation and similarity measures are widely used in all the areas of sciences and social sciences often the variables are not numbers but are instead qualitative descriptors called categorical data we define and study similarity matrix as a measure of similarity for the case of categorical data this is of interest due to a deluge of categorical data such as movie ratings top 10 rankings and data from social media in the public domain that require analysis we show that the statistical properties of the spectra of similarity matrices constructed from categorical data follow random matrix predictions with the dominant eigenvalue being an exception we demonstrate this approach by applying it to the data for indian general elections and sea level pressures in the north atlantic ocean

localized short range correlations in the spectrum of the equal time correlation matrix we suggest a procedure to identify those parts of the spectrum of the equal time correlation matrix mathbf c where relevant information about correlations of a multivariate time series is induced using an ensemble average over each of the distances between eigenvalues all nearest neighbor distributions can be calculated individually we present numerical examples where a information about cross correlations is found in the so called bulk of eigenvalues which generally is thought to contain only random correlations and where b the information extracted from the lower edge of the spectrum of mathbf c is statistically more significant than that extracted from the upper edge we apply the analysis to electroencephalographic recordings with epileptic events

the shannon entropy of sudoku matrices we study properties of an ensemble of sudoku matrices a special type of doubly stochastic matrix when normalized using their statistically averaged singular values the determinants are very near

patterns of a spatial exploration under time evolution of the attractiveness persistent nodes degree distribution and spectral properties abstract this work explores the features of a graph generated by agents that hop from one node to another node where the nodes have evolutionary attractiveness the jumps are governed by boltzmann like transition probabilities that depend both on the euclidean distance between the nodes and on the ratio of the attractiveness between them it is shown that persistent nodes i e nodes that never been reached by this special random walk are possible in the stationary limit differently from the case where the attractiveness is fixed and equal to one for all nodes 1 simultaneously one also investigates the spectral properties and statistics related to the attractiveness and degree distribution of the evolutionary network finally a study of the crossover between persistent phase and no persistent phase was performed and it was also observed the existence of a special type of transition probability which leads to a power law behaviour for the time evolution of the persistence

econophysics financial time series from a statistical physics point of view in recent years physicists have started applying concepts and methods of statistical physics to study economic problems the word econophysics is sometimes used to refer to this work much recent work is focused on understanding the statistical properties of nancial time series one reason for this interest is that nancial markets are examples of complex interacting systems for which a huge amount of data exist and it is possible that nancial time series viewed from a dierent perspective might yield new results this article reviews the results of three recent phenomenological studies i the probability distribution of stock price uctuations stock price uctuations occur in all magnitudes in analogy to earthquakes from tiny uctuations to drastic events such as market crashes the distribution of price uctuations decays with a power law tail well outside the levy stable regime and describes uctuations that dier by as much as eight orders of magnitude in addition this distribution preserves its functional form for uctuations on time scales that dier by three orders of magnitude from 1 min up to approximately 10 d ii correlations in nancial time series while price uctuations themselves have rapidly decaying correlations the magnitude of uctuations measured by either the absolute value or the square of the price uctuations has correlations that decay as a power law and persist for several months iii correlations among dierent companies the third result bears on the application of random matrix theory to understand the correlations among price uctuations of any two dierent stocks from a study of the eigenvalue statistics of the cross correlation matrix constructed from price uctuations of the leading 1000 stocks we nd that the largest 1 of the eigenvalues and the corresponding eigenvectors show systematic deviations from the predictions for a random matrix whereas the rest of the eigenvalues conform to random matrix behavior suggesting that these 1 of the eigenvalues contain system specic information about correlated time evolution of dierent companies c 2000 published by elsevier science b v all rights reserved

convergence of a splitting method of high order for reaction diffusion systems in this article we prove the convergence of a splitting scheme of high order for a reaction diffusion system of the form u t m u f u 0 where m is an m m matrix whose spectrum is included in rz 0 this scheme is obtained by applying a richardson extrapolation to a strang formula

a random matrix theory approach to financial cross correlations it is common knowledge that any two rms in the economy are correlated even rms belonging to dierent sectors of an industry may be correlated because of indirect correlations how can we analyze and understand these correlations this article reviews recent results regarding cross correlations between stocks specically we use methods of random matrix theory rmt which originated from the need to understand the interactions between the constituent elements of complex interacting systems to analyze the cross correlation matrix c of returns we analyze 30 min returns of the largest 1000 us stocks for the 2 year period 1994 1995 we nd that the statistics of approximately 20 of the largest eigenvalues 2 show deviations from the predictions of rmt to test that the rest of the eigenvalues are genuinely random we test for universal properties such as eigenvalue spacings and eigenvalue correlations and demonstrate that c shares universal properties with the gaussian orthogonal ensemble of random matrices the statistics of the eigenvectors of c conrm the deviations of the largest few eigenvalues from the rmt prediction we also nd that these deviating eigenvectors are stable in time in addition we quantify the number of rms that participate signicantly to an eigenvector using the concept of inverse participation ratio borrowed from localization theory c 2000 published by elsevier science b v all rights reserved

spectral distribution of random matrices from binary linear block codes let c be a binary linear block code of length n dimension k and minimum hamming distance d over gf 2 n let d denote the minimum hamming distance of the dual code of c over gf 2 n let e gf 2 n 1 1 n be the component wise mapping e vi 1 vi for v v1 v2 vn gf 2 n finally for p n let mmb c be a p n random matrix whose rows are obtained by mapping a uniformly drawn set of size p of the codewords of c under e it is shown that for d large enough and y p n 0 1 fixed as n the empirical spectral distribution of the gram matrix of 1 n mmb c resembles that of a random i i d rademacher matrix i e the marchenko pastur distribution moreover an explicit asymptotic uniform bound on the distance of the empirical spectral distribution of the gram matrix of 1 n mmb c to the marchenko pastur distribution as a function of y and d is presented

similarities and differences between physics and economics in this opening talk we discuss some of the similarities between work being done by economists and by physicists seeking to contribute to economics we also mention some of the dierences in the approaches taken and justify these dierent approaches by developing the argument that by approaching the same problem from dierent points of view new results might emerge in particular we review some recent results for example the 1nding that there are two new universal scaling models in economics i the 3uctuation of price changes of any stock market is characterized by a pdf which is a simple power law with exponent 4 that extends over 10 2 standard deviations a factor of 10 8 on the y axis ii for a wide range of economic organizations the histogram that shows how size of organization is inversely correlated to 3uc tuations in size with an exponent 1 6 neither of these two new laws has a 1rm theoretical foundation we also discuss results that are reminiscent of phase transitions in spin systems where the divergent behavior of the response function at the critical point zero magnetic 1eld leads to large 3uctuations c 2001 published by elsevier science b v

correlation and network analysis of global financial indices random matrix theory rmt and network methods are applied to investigate the correlation and network properties of 20 financial indices the results are compared before and during the financial crisis of 2008 in the rmt method the components of eigenvectors corresponding to the second largest eigenvalue form two clusters of indices in the positive and negative directions the components of these two clusters switch in opposite directions during the crisis the network analysis uses the fruchterman reingold layout to find clusters in the network of indices at different thresholds at a threshold of 0 6 before the crisis financial indices corresponding to the americas europe and asia pacific form separate clusters on the other hand during the crisis at the same threshold the american and european indices combine together to form a strongly linked cluster while the asia pacific indices form a separate weakly linked cluster if the value of the threshold is further increased to 0 9 then the european indices france germany and the united kingdom are found to be the most tightly linked indices the structure of the minimum spanning tree of financial indices is more starlike before the crisis and it changes to become more chainlike during the crisis the average linkage hierarchical clustering algorithm is used to find a clearer cluster structure in the network of financial indices the cophenetic correlation coefficients are calculated and found to increase significantly which indicates that the hierarchy increases during the financial crisis these results show that there is substantial change in the structure of the organization of financial indices during a financial crisis

collective behavior of cryptocurrency price changes abstract digital assets termed cryptocurrencies are correlated we analyze cross correlations between price changes of different cryptocurrencies using methods of random matrix theory and minimum spanning trees we find that the cross correlation matrix exhibits non trivial hierarchical structures and groupings of cryptocurrency pairs which are not present in the partial cross correlations in sharp contrast to the predictions for other financial markets we discover that most of the eigenvalues in the spectrum of the cross correlation matrix do not agree with the universal predictions of random matrix theory but the few of the largest eigenvalues deviate as expected the minimum spanning tree of cryptocurrency cross correlations reveals distinct community structures that are surprisingly stable collective behaviors that are present in the cryptocurrency market can be useful for the construction of portfolio of cryptocurrencies as well as for future research on the subject

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

detection and characterization of changes of the correlation structure in multivariate time series we propose a method based on the equal time correlation matrix as a sensitive detector for phase shape correlations in multivariate data sets the key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix consequently detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors the performance of the technique is demonstrated by application to n f tori autoregressive models and coupled chaotic systems the high sensitivity the comparatively small computational effort and the excellent time resolution of the method recommend it for application to the analysis of complex spatially extended nonstationary systems

collective origin of the coexistence of apparent random matrix theory noise and of factors in large sample correlation matrices abstract through simple analytical calculations and numerical simulations we demonstrate the generic existence of a self organized macroscopic state in any large multivariate system possessing non vanishing average correlations between a finite fraction of all pairs of elements the coexistence of an eigenvalue spectrum predicted by random matrix theory rmt and a few very large eigenvalues in large empirical correlation matrices is shown to result from a bottom up collective effect of the underlying time series rather than a top down impact of factors our results in excellent agreement with previous results obtained on large financial correlation matrices show that there is relevant information also in the bulk of the eigenvalue spectrum and rationalize the presence of market factors previously introduced in an ad hoc manner

global solutions to the 3 d incompressible inhomogeneous navier stokes system in this paper we consider the global well posedness of the 3 d incompressible inhomogeneous navier stokes equations with initial data in the critical besov spaces a0 bq 13q r3 u0 u0h u03 bp 1 1 3p r3 for p q satisfying 1 q p 6 and 1q 1p 13 more precisely we prove that there exist two positive constants c0 c0 such that if a0 bq 13q u0h bp 1 1 3p exp c0 u03 bp 1 1 3p2 2 c0 then 1 3 has a unique global solution a l r bq 13q r3 u l r bp 1 1 3p r3 l1 r bp 11 3p r3 in particular this result implies the global well posedness result in abidi and paicu 2007 2 for the inhomogeneous navier stokes system with small initial data

global solutions to the 3 d incompressible anisotropic navier stokes system in the critical spaces in this paper we consider the global wellposedness of the 3 d incompressible anisotropic navier stokes equations with initial data in the critical besov sobolev type spaces mathcal b and mathcal b frac12 frac12 4 see definitions 1 1 and 1 2 below in particular we proved that there exists a positive constant c such that ans has a unique global solution with initial data u 0 u 0 h u 0 3 which satisfies u 0 h mathcal b exp bigl frac c nu 4 u 0 3 mathcal b 4 bigr leq c 0 nu or u 0 h mathcal b frac12 frac12 4 exp bigl frac c nu 4 u 0 3 mathcal b frac12 frac12 4 4 bigr leq c 0 nu for some c 0 sufficiently small to overcome the difficulty that gronwall s inequality can not be applied in the framework of chemin lerner type spaces widetilde l p t mathcal b we introduced here sort of weighted chemin lerner type spaces widetilde l 2 t f mathcal b for some apropriate l 1 function f t

on the well posedness of 2 d incompressible navier stokes equations with variable viscosity in critical spaces abstract in this paper we first prove the local well posedness of the 2 d incompressible navier stokes equations with variable viscosity in critical besov spaces with negative regularity indices without smallness assumption on the variation of the density the key is to prove for p 1 4 and a b p 1 2 p r 2 that the solution mapping h a f to the 2 d elliptic equation div 1 a div f is bounded on b p 1 2 p 1 r 2 more precisely we prove that b p 1 2 p 1 c 1 a b p 1 2 p 2 f b p 1 2 p 1 the proof of the uniqueness of solution to 1 2 relies on a lagrangian approach 15 16 17 when the viscosity coefficient is a positive constant we prove that 1 2 is globally well posed

erratum to global wellposed problem for the 3 d incompressible anisotropic navier stokes equations in an anisotropic space in this paper we consider a global wellposed problem for the 3 d incompressible anisotropic navier stokes equations ans we prove the global wellposedness for ans provided the initial horizontal data are sufficient small in the scaling invariant besov sobolev type space b 0 frac 1 2 in particular the result implies the global wellposedness of ans with large initial vertical velocity

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

quantifying economic fluctuations abstract this manuscript is a brief summary of a talk designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena scale invariance and universality can be useful in guiding research on interpreting empirical data on economic fluctuations using this conceptual framework as a guide we empirically quantify the relation between trading activity measured by the number of transactions n and the price change g t for a given stock over a time interval t t t we relate the time dependent standard deviation of price changes volatility to two microscopic quantities the number of transactions n t in t and the variance w2 t of the price changes for all transactions in t we find that the long ranged volatility correlations are largely due to those of n we then argue that the tail exponent of the distribution of n is insufficient to account for the tail exponent of p g x since n and w display only weak inter dependency our results show that the fat tails of the distribution p g x arises from w finally we review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory rmt rmt makes predictions for universal properties that do not depend on the interactions between the elements comprising the system and deviations from rmt provide clues regarding system specific properties we compare the statistics of the cross correlation matrix c whose elements cij are the correlation coefficients of price fluctuations of stock 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 we also discuss results that are reminiscent of phase transitions in spin systems where the divergent behavior of the response function at the critical point zero magnetic field leads to large fluctuations and we discuss a curious symmetry breaking a feature qualitatively identical to the behavior of the probability density of the magnetization for fixed values of the inverse temperature

identifying complexity by means of matrices abstract complexity is an interdisciplinary concept which first of all addresses the question of how order emerges out of randomness for many reasons matrices provide a very practical and powerful tool in approaching and quantifying the related characteristics based on several natural complex dynamical systems like the strongly interacting quantum many body systems the human brain and the financial markets by relating empirical observations to the random matrix theory and quantifying deviations in terms of a reduced dimensionality we present arguments in favour of the statement that complexity is a phenomenon at the edge between collectivity and chaos

random matrix theory for portfolio optimization a stability approach we apply random matrix theory rmt to an empirically measured financial correlation matrix c and show that this matrix contains a large amount of noise in order to determine the sensitivity of the spectral properties of a random matrix to noise we simulate a set of data and add different volumes of random noise having ascertained that the eigenspectrum is independent of the standard deviation of added noise we use rmt to determine the noise percentage in a correlation matrix based on real data from s p500 eigenvalue and eigenvector analyses are applied and the experimental results for each of them are presented to identify qualitatively and quantitatively different spectral properties of the empirical correlation matrix to a random counterpart finally we attempt to separate the noisy part from the non noisy part of c we apply an existing technique to cleaning c and then discuss its associated problems we propose a technique of filtering c that has many advantages from the stability point of view over the existing method of cleaning

inhomogeneous navier stokes equations in the half space with only bounded density abstract in this paper we establish the global existence and uniqueness of solutions to the inhomogeneous navier stokes system in the half space the initial density only has to be bounded and close enough to a positive constant the initial velocity belongs to some critical besov space and the l norm of the inhomogeneity plus the critical norm to the horizontal components of the initial velocity has to be very small compared to the exponential of the norm to the vertical component of the initial velocity with a little bit more regularity for the initial velocity those solutions are proved to be unique in the last section of the paper our results are partially extended to the bounded domain case

cross correlations in an emerging market financial data abstract we report the results of a study of the spectral properties of the correlation matrix of price fluctuations in the stock prices in an emerging market istanbul stock exchange market by using random matrix theory although properties such as the distributions of correlation matrix elements eigenvalue spacings and components of the eigenvectors show similar trends to those observed for the other markets the eigenvalue distribution is found to be markedly different compared to similar studies

spectral density of a wishart model for nonsymmetric correlation matrices the wishart model for real symmetric correlation matrices is defined as mathsf w mathsf aa t where matrix mathsf a is usually a rectangular gaussian random matrix and mathsf a t is the transpose of mathsf a analogously for nonsymmetric correlation matrices a model may be defined for two statistically equivalent but different matrices mathsf a and mathsf b as mathsf ab t the corresponding wishart model thus is defined as mathbf c mathsf ab t mathsf ba t we study the spectral density of mathbf c for the case when mathsf a and mathsf b are not statistically independent the ensemble average of such nonsymmetric matrices therefore does not simply vanishes to a null matrix in this paper we derive a pastur self consistent equation which describes spectral density of mathbf c at large matrix dimension we complement our analytic results with numerics

rational decisions random matrices and spin glasses we consider the problem of rational decision making in the presence of nonlinear constraints by using tools borrowed from spin glass and random matrix theory we focus on the portfolio optimisation problem we show that the number of optimal solutions is generally exponentially large and each of them is fragile rationality is in this case of limited use in addition this problem is related to spin glasses with levy like long ranged couplings for which we show that the ground state is not exponentially degenerate

an analysis of cross correlations in an emerging market abstract we apply random matrix theory to compare correlation matrix estimators c obtained from emerging market data the correlation matrices are constructed from 10 years of daily data for stocks listed on the johannesburg stock exchange jse from january 1993 to december 2002 we test the spectral properties of c against random matrix predictions and find some agreement between the distributions of eigenvalues nearest neighbour spacings distributions of eigenvector components and the inverse participation ratios for eigenvectors we show that interpolating both missing data and illiquid trading days with a zero order hold increases agreement with rmt predictions for the more realistic estimation of correlations in an emerging market we suggest a pairwise measured data correlation matrix for the data set used this approach suggests greater temporal stability for the leading eigenvectors an interpretation of eigenvectors in terms of trading strategies is given as opposed to classification by economic sectors

sparls the sparse rls algorithm we develop a recursive l1 regularized least squares sparls algorithm for the estimation of a sparse tap weight vector in the adaptive filtering setting the sparls algorithm exploits noisy observations of the tap weight vector output stream and produces its estimate using an expectation maximization type algorithm we prove the convergence of the sparls algorithm to a near optimal estimate in a stationary environment and present analytical results for the steady state error simulation studies in the context of channel estimation employing multipath wireless channels show that the sparls algorithm has significant improvement over the conventional widely used recursive least squares rls algorithm in terms of mean squared error mse moreover these simulation studies suggest that the sparls algorithm with slight modifications can operate with lower computational requirements than the rls algorithm when applied to tap weight vectors with fixed support

incompressible flows with piecewise constant density we investigate the incompressible navier stokes equations with variable density the aim is to prove existence and uniqueness results in the case of discontinuous initial density in dimension n 2 3 assuming only that the initial density is bounded and bounded away from zero and that the initial velocity is smooth enough we get the local in time existence of unique solutions uniqueness holds in any dimension and for a wider class of velocity fields in particular all those results are true for piecewise constant densities with arbitrarily large jumps global results are established in dimension two if the density is close enough to a positive constant and in n dimensions if in addition the initial velocity is small the lagrangian formulation for describing the flow plays a key role in the analysis that is proposed in the present paper

eigenvalue density in hermitian matrix models by the lax pair method in this paper a new method is discussed to derive the eigenvalue density in a hermitian matrix model with a general potential the density is considered on one interval or multiple disjoint intervals the method is based on lax pair theory and the cayley hamilton theorem by studying the orthogonal polynomials associated with the hermitian matrix model it is obtained that the restriction conditions for the parameters in the density are connected to the discrete painleve i equation and the results are related to the scalar riemann hilbert problem some special density functions are also discussed in association with the known results in this subject

global fujita kato solution of 3 d inhomogeneous incompressible navier stokes system abstract in this paper we shall prove the global existence of weak solutions to 3d inhomogeneous incompressible navier stokes system ins with initial density in the bounded function space and having a positive lower bound and with initial velocity being sufficiently small in the critical besov space b 2 1 1 2 this result corresponds to the fujita kato solutions of the classical navier stokes system the same idea can be used to prove the global existence of weak solutions in the critical functional framework to ins with one component of the initial velocity being large and can also be applied to provide a lower bound for the lifespan of smooth enough solutions of ins