quadratic operator relations and bethe equations for spin 1 2 richardson gaudin models in this work we demonstrate how one can in a generic approach derive a set of n simple quadratic bethe equations for integrable richardson gaudin rg models built out of n spins 1 2 these equations depend only on the n eigenvalues of the various conserved charges so that any solution of these equations defines indirectly through the corresponding set of eigenvalues one particular eigenstate the proposed construction covers the full class of integrable rg models of the xyz including the subclasses of xxz and xxx models type realised in terms of spins 1 2 coupled with one another through terms including as well magnetic field like terms linear in the pauli matrices the approach exclusively requires integrability defined here only by the requirement that n conserved charges r i with such that exist the result is therefore valid and equally simple for models with or without symmetry with or without a properly defined pseudo vacuum as well as for models with non skew symmetric couplings

on rational solutions of yang baxter equations maximal orders in loop algebra in 1982 belavin and drinfeld listed all elliptic and trigonometric solutionsx u v of the classical yang baxter equation cybe wherex takes values in a simple complex lie algebrag and left the classification problem of the rational one open in 1984 drinfeld conjectured that if a rational solution is equivalent to a solution of the formx u v c2 u v r u v wherec2 is the quadratic casimir element andr is a polynomial inu v then deg u r deg v r 1 in another paper i proved this conjecture forg sl n and reduced the problem of listing nontrivial i e nonequivalent toc2 u v solutions of cybe to classification of quasi frobenius subalgebras of g they in turn are related with the so called maximal orders in the loop algebra of g corresponding to the vertices of the extended dynkin diagramd e g in this paper i give an algorithm which enables one to list all solutions and illustrate it with solutions corresponding to vertices ofd e g with coefficient 2 or 3 in particular i will find all solutions forg o 5 and some solutions forg o 7 o 10 o 14 andg2

bayesian robust tensor factorization for incomplete multiway data we propose a generative model for robust tensor factorization in the presence of both missing data and outliers the objective is to explicitly infer the underlying low candecomp parafac cp rank tensor capturing the global information and a sparse tensor capturing the local information also considered as outliers thus providing the robust predictive distribution over missing entries the low cp rank tensor is modeled by multilinear interactions between multiple latent factors on which the column sparsity is enforced by a hierarchical prior while the sparse tensor is modeled by a hierarchical view of student t distribution that associates an individual hyperparameter with each element independently for model learning we develop an efficient variational inference under a fully bayesian treatment which can effectively prevent the overfitting problem and scales linearly with data size in contrast to existing related works our method can perform model selection automatically and implicitly without the need of tuning parameters more specifically it can discover the groundtruth of cp rank and automatically adapt the sparsity inducing priors to various types of outliers in addition the tradeoff between the low rank approximation and the sparse representation can be optimized in the sense of maximum model evidence the extensive experiments and comparisons with many state of the art algorithms on both synthetic and real world data sets demonstrate the superiorities of our method from several perspectives

rank regularization and bayesian inference for tensor completion and extrapolation a novel regularizer of the parafac decomposition factors capturing the tensor s rank is proposed in this paper as the key enabler for completion of three way data arrays with missing entries set in a bayesian framework the tensor completion method incorporates prior information to enhance its smoothing and prediction capabilities this probabilistic approach can naturally accommodate general models for the data distribution lending itself to various fitting criteria that yield optimum estimates in the maximum a posteriori sense in particular two algorithms are devised for gaussian and poisson distributed data that minimize the rank regularized least squares error and kullback leibler divergence respectively the proposed technique is able to recover the ground truth tensor rank when tested on synthetic data and to complete brain imaging and yeast gene expression datasets with 50 and 15 of missing entries respectively resulting in recovery errors at 11 db and 15 db

quantum integrable systems non skew symmetric r matrices and algebraic bethe ansatz we prove the integrability of the general quantum hamiltonian systems governed by an arbitrary non skew symmetric so 3 valued nondynamical classical r matrix with spectral parameters we consider the most interesting example of these quantum integrable systems namely the so 3 generalized gaudin systems in detail in the case of an arbitrary r matrix which is diagonal in the sl 2 basis we calculate the spectrum and the eigenvalues of the corresponding hamiltonians using the algebraic bethe ansatz technique

smooth parafac decomposition for tensor completion in recent years low rank based tensor completion which is a higher order extension of matrix completion has received considerable attention however the low rank assumption is not sufficient for the recovery of visual data such as color and 3d images when the ratio of missing data is extremely high in this paper we consider smoothness constraints as well as low rank approximations and propose an efficient algorithm for performing tensor completion that is particularly powerful regarding visual data the proposed method admits significant advantages owing to the integration of smooth parafac decomposition for incomplete tensors and the efficient selection of models in order to minimize the tensor rank thus our proposed method is termed as smooth parafac tensor completion spc in order to impose the smoothness constraints we employ two strategies total variation spc tv and quadratic variation spc qv and invoke the corresponding algorithms for model learning extensive experimental evaluations on both synthetic and real world visual data illustrate the significant improvements of our method in terms of both prediction performance and efficiency compared with many state of the art tensor completion methods

isomonodromic deformations generalized knizhnik zamolodchikov equations and non skew symmetric classical r matrices using non skew symmetric classical r matrices r12 u v with spectral parameters we generalize fuchsian systems and schlesinger equations isomonodromy equations viewing the last as the nonautonomous hamiltonian equations corresponding to the generalized gaudin hamiltonians quantizing these systems we construct generalization of knizhnik zamolodchikov equations corresponding to the non skew symmetric classical r matrices r12 u v with spectral parameters in the case of an ordinary skew symmetric classical r matrix r12 u v depending on the difference of the spectral parameters we reobtain as special partial cases a standard generalization of fuchsian systems schlesinger and knizhnik zamolodchikov equations

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

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

classical yang baxter equations and quantum integrable systems quantum integrable models associated with nondegenerate solutions of classical yang baxter equations related to the simple lie algebras are investigated these models are diagonalized for rational and trigonometric solutions in the cases of sl n gl n o n and sp n algebras the analogy with the quantum inverse scattering method is demonstrated

statistical properties of the volatility of price fluctuations we study the statistical properties of volatility measured by locally averaging over a time window t the absolute value of price changes over a short time interval ensuremath delta t we analyze the s 500 stock index for the 13 year period jan 1984 to dec 1996 we find that the cumulative distribution of the volatility is consistent with a power law asymptotic behavior characterized by an exponent ensuremath mu ensuremath approx 3 similar to what is found for the distribution of price changes the volatility distribution retains the same functional form for a range of values of t further we study the volatility correlations by using the power spectrum analysis both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents also both methods show the presence of a crossover at approximately 1 5 days in addition we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 u s companies over the two year period jan 1994 to dec 1995

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

fractionally integrated generalized autoregressive conditional heteroskedasticity abstract the new class of fractionally integrated generalized autoregressive conditionally heteroskedastic figarch processes is introduced the conditional variance of the process implies a slow hyperbolic rate of decay for the influence of lagged squared innovations unlike i d processes for the mean maximum likelihood estimates mle of the figarch parameters are argued to be t 1 2 consistent the small sample behavior of an approximate mle procedure is assessed through a simulation study which also documents how the estimation of a standard garch model tends to produce integrated or igarch like estimates an empirical example with daily deutschmark u s dollar exchange rates illustrates the practical relevance of the new figarch specification

long term memory in stock market prices a test for long term memory that is robust to short range dependence is developed it is a modification of the r s statistic and the relevant asymptotic sampling theory is derived via functional central limit theory contrary to previous findings when applied to daily and monthly stock returns indexes over several time periods this test yields no evidence of long range dependence once short range dependence is accounted for monte carlo experiments show that the modified r s test has power against at least two specific models of long term memory suggesting that models with short range dependence may adequately capture the behavior of historical stock returns copyright 1991 by the econometric society

simultaneous tensor decomposition and completion using factor priors the success of research on matrix completion is evident in a variety of real world applications tensor completion which is a high order extension of matrix completion has also generated a great deal of research interest in recent years given a tensor with incomplete entries existing methods use either factorization or completion schemes to recover the missing parts however as the number of missing entries increases factorization schemes may overfit the model because of incorrectly predefined ranks while completion schemes may fail to interpret the model factors in this paper we introduce a novel concept complete the missing entries and simultaneously capture the underlying model structure to this end we propose a method called simultaneous tensor decomposition and completion stdc that combines a rank minimization technique with tucker model decomposition moreover as the model structure is implicitly included in the tucker model we use factor priors which are usually known a priori in real world tensor objects to characterize the underlying joint manifold drawn from the model factors by exploiting this auxiliary information our method leverages two classic schemes and accurately estimates the model factors and missing entries we conducted experiments to empirically verify the convergence of our algorithm on synthetic data and evaluate its effectiveness on various kinds of real world data the results demonstrate the efficacy of the proposed method and its potential usage in tensor based applications it also outperforms state of the art methods on multilinear model analysis and visual data completion tasks

fractionally integrated process for transition economics we analyze the european transition economics and show that many time series of major indices exhibit i power law correlations in their values ii power law correlations in their magnitudes and iii an asymmetric probability distribution applying the phase randomization procedure to these time series we show that magnitude correlations completely vanish we propose a stochastic model that can generate time series with features i ii and iii and we show by means of numerical simulations that this model is capable of reproducing these three features found in the empirical data

varieties of long memory models long memory is defined as a series having a slowly declining correlogram or equivalently an infinite spectrum at zero frequency fractional integrated processes have such properties but here it is pointed out that a number of other processes can also be long memory including generalized fractionally integrated models arising from aggregation time changing coefficient models and possibly nonlinear models it seems that there are many classes of processes that deserve further study the relevance of long memory is illustrated using absolute returns from a daily stock market index

generalized gaudin spin chains nonskew symmetric r matrices and reflection equation algebras using a one parametric family of automorphisms u of finite dimensional simple lie algebras g and classical skew symmetric r matrices r12 u v we construct twisted nonskew symmetric classical r matrices r12 u v with spectral parameters we show that in the special case u adk u and classical matrix lie algebras g they are connected with the quasiclassical limit of the reflection equation algebras using them and results of our previous papers we construct new integrable quantum spin chains of the generalized gaudin type with an arbitrary value of spin living at each site

modeling persistence in hydrological time series using fractional differencing the class of autoregressive integrated moving average arima time series models may be generalized by permitting the degree of differencing d to take fractional values models including fractional differencing are capable of representing persistent series d 0 or short memory series d 0 the class of fractionally differenced arima processes provides a more flexible way than has hitherto been available of simultaneously modeling the long term and short term behavior of a time series in this paper some fundamental properties of fractionally differenced arima processes are presented methods of simulating these processes are described estimation of the parameters of fractionally differenced arima models is discussed and an approximate maximum likelihood method is proposed the methodology is illustrated by fitting fractionally differenced models to time series of streamflows and annual temperatures

fast algorithms for multidimensional dct to dct computation between a block and its associated subblocks in this paper we first propose an efficient algorithm for computing one dimensional 1 d discrete cosine transform dct for a signal block given its two adjacent subblocks in the dct domain and then introduce several algorithms for the fast computation of multidimensional m d dct with size n sub 1 spl times n sub 2 spl times spl times n sub m given 2 sup m subblocks of dct coefficients with size n sub 1 2 spl times n sub 2 2 spl times spl times n sub m 2 where n sub i i 1 2 m are powers of 2 obviously the row column method which employs the most efficient algorithms along each dimension reduces the computational complexity considerably compared with the traditional method which employs only the one dimensional 1 d fast dct and inverse dct idct algorithms however when m spl ges 2 the traditional method which employs the most efficient multidimensional dct idct algorithms has lower computational complexity than the row column method besides we propose a direct method by dividing the data into 2 sup m parts for independent fast computation in which only two steps of r dimensional r 1 2 m idct and additional multiplications and additions are required if all the dimensional sizes are the same the number of multiplications required for the direct method is only 2 sup m 1 m2 sup m 1 times of that required for the row column method and if n spl ges 2 sup 2m 1 the computational efficiency of the direct method is surely superior to that of the traditional method which employs the most efficient multidimensional dct idct algorithms

salient features of dependence in daily us stock market indices abstract this paper deals with the analysis of long range dependence in the us stock market we focus first on the log values of the dow jones industrial average standard and poors 500 and nasdaq indices daily from february 1971 to february 2007 the volatility processes are examined based on the squared and the absolute values of the returns series and the stability of the parameters across time is also investigated in both the level and the volatility processes a method that permits us to estimate fractional differencing parameters in the context of structural breaks is conducted in this paper finally the day of the week effect is examined by looking at the order of integration for each day of the week providing also a new modeling approach to describe the dependence in this context

many poled r matrix lie algebras lax operators and integrable systems for each finite dimensional simple lie algebra g starting from a general g g valued solution r u v of the generalized non dynamical classical yang baxter equation we construct n poled infinite dimensional lie algebras gr n of g valued meromorphic functions with the poles in a fixed set of points 1 n we apply the constructed algebras to the theory of finite dimensional integrable systems and theory of soliton equations and obtain with their help the most general form of anisotropic chiral field type equations as well as the most general form of integrable n top systems and integrable cases of n interacting kirchhoff systems

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

tensor train decomposition a simple nonrecursive form of the tensor decomposition in d dimensions is presented it does not inherently suffer from the curse of dimensionality it has asymptotically the same number of parameters as the canonical decomposition but it is stable and its computation is based on low rank approximation of auxiliary unfolding matrices the new form gives a clear and convenient way to implement all basic operations efficiently a fast rounding procedure is presented as well as basic linear algebra operations examples showing the benefits of the decomposition are given and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19 dimensional operator

a fused cp factorization method for incomplete tensors low rank tensor completion methods have been advanced recently for modeling sparsely observed data with a multimode structure however low rank priors may fail to interpret the model factors of general tensor objects the most common method to address this drawback is to use regularizations together with the low rank priors however due to the complex nature and diverse characteristics of real world multiway data the use of a single or a few regularizations remains far from efficient and there are limited systematic experimental reports on the advantages of these regularizations for tensor completion to fill these gaps we propose a modified cp tensor factorization framework that fuses the l 2 norm constraint sparseness l 1 norm manifold and smooth information simultaneously the factorization problem is addressed through a combination of nesterov s optimal gradient descent method and block coordinate descent here we construct a smooth approximation to the l 1 norm and tv norm regularizations and then the tensor factor is updated using the projected gradient method where the step size is determined by the lipschitz constant extensive experiments on simulation data visual data completion intelligent transportation systems and gps data of user involvement are conducted and the efficiency of our method is confirmed by the results moreover the obtained results reveal the characteristics of these commonly used regularizations for tensor completion in a certain sense and give experimental guidance concerning how to use them