fast isodata clustering algorithms abstract the computational requirements of any clustering method are identified as the major bottleneck in the effective exploratory data analysis task partial sum and nearest neighbouring distance methods are proposed to speed up the k means clustering algorithm with euclidean distance norm use of the mahalanobis distance norm instead of the euclidean distance norm is identified for the effective exploration of the real structure of the data lower triangular matrix and expanded distance function approaches are proposed to speed up the mahalanobis distance based k means algorithm further use of the dynamic clustering method for optimal global partition of the data is identified and use of the lower triangular matrix approach to speed up dynamic clustering is proposed other methods to speed up the dynamic clustering process are identified use of the image auto correlation property is identified for further speeding up of the proposed methods in the clustering of image data

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

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

a survey of the theory of source coding with a fidelity criterion the purpose of this paper is threefold 1 to acquaint the reader with the types of problems that have been considered in the area of source coding with a fidelity criterion 2 to survey results that have been obtained on these problems and 3 to outline future research trends in the area

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

model selection and akaike s information criterion aic the general theory and its analytical extensions during the last fifteen years akaike s entropy based information criterion aic has had a fundamental impact in statistical model evaluation problems this paper studies the general theory of the aic procedure and provides its analytical extensions in two ways without violating akaike s main principles these extensions make aic asymptotically consistent and penalize overparameterization more stringently to pick only the simplest of the true models these selection criteria are called caic and caicf asymptotic properties of aic and its extensions are investigated and empirical performances of these criteria are studied in choosing the correct degree of a polynomial model in two different monte carlo experiments under different conditions

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

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

efficient algorithms for universal portfolios a constant rebalanced portfolio is an investment strategy that keeps the same distribution of wealth among a set of stocks from day to day there has been much work on cover s universal algorithm which is competitive with the best constant rebalanced portfolio determined in hindsight cover 1991 helmbold et al 1998 blum and kalai 1999 foster and vohra 1999 vovk 1998 cover and ordentlich 1996a cover 1996c while this algorithm has good performance guarantees all known implementations are exponential in the number of stocks restricting the number of stocks used in experiments helmbold et al 1998 cover and ordentlich 1996a ordentlich and cover 1996b cover 1996c blum and kalai 1999 we present an efficient implementation of the universal algorithm that is based on non uniform random walks that are rapidly mixing applegate and kannan 1991 lovasz and simonovits 1992 frieze and kannan 1999 this same implementation also works for non financial applications of the universal algorithm such as data compression cover 1996c and language modeling chen et al 1999

fast and efficient coding of information sources the author considers the problem of source coding and investigates the cases of known and unknown statistics the efficiency of the compression codes can be estimated by three characteristics 1 the redundancy r defined as the maximal difference between the average codeword length and shannon entropy in case the letters are generated by a bernoulli source 2 the size in bits of the encoder and the encoder programs s when implemented on a computer and 3 the average time required for encoding and decoding of a single letter t he investigates s and t as a function of r when r spl rarr 0 all known methods may be divided into two classes the ziv lempel codes and their variants fall under the first class and the arithmetic code with the lynch davisson code fall under the second one the codes from the first class need exponential memory size s 0 exp 1 r for redundancy r when r spl rarr o the methods from the second class have a small memory size but a low encoding speed s 0 1 r sup const t 0 log sup const 1 r log log 1 r in this paper the author presents a code that combines the merits of both classes the memory size is small and the speed is high s 0 1 r sup const t 0 log sup const 1 r log log 1 r

coding of sources with two sided geometric distributions and unknown parameters lossless compression is studied for a countably infinite alphabet source with an unknown off centered two sided geometric tsg distribution which is a commonly used statistical model for image prediction residuals we demonstrate that arithmetic coding based on a simple strategy of model adaptation essentially attains the theoretical lower bound to the universal coding redundancy associated with this model we then focus on more practical codes for the tsg model that operate on a symbol by symbol basis and study the problem of adaptively selecting a code from a given discrete family by taking advantage of the structure of the optimum huffman tree for a known tsg distribution which enables simple calculation of the codeword of every given source symbol an efficient adaptive strategy is derived

on the well posedness of the full compressible navier stokes system in critical besov spaces abstract we are concerned with the cauchy problem of the full compressible navier stokes equations satisfied by viscous and heat conducting fluids in r n we focus on the so called critical besov regularity framework in this setting it is natural to consider initial densities 0 velocity fields u 0 and temperatures 0 with a 0 0 1 b p 1 n p u 0 b p 1 n p 1 and 0 b p 1 n p 2 after recasting the whole system in lagrangian coordinates and working with the total energy along the flow rather than with the temperature we discover that the system may be solved by means of banach fixed point theorem in a critical functional framework whenever the space dimension is n 2 and 1 p 2 n back to eulerian coordinates this allows to improve the range of p s for which the system is locally well posed compared to 7

supervising isodata with an information theoretic stopping rule abstract new biomedical imaging modalities such as magnetic resonance imaging mri provide fertile multidimensional environments for the automatic identification of biological soft tissues but lack the a priori information required to appropriately train supervised classifiers hierarchical cluster analysis techniques can generate this information but they are inefficient on large data sets we have developed an unsupervised clustering method that is a variant of the well known isodata algorithm we replaced the heuristic rules that control isodata with rules that search for the minimum value of an information theoretic criterion the criteria investigated in this study are akaike s information criterion aic and the consistent aic caic both measure the global fit of a cluster model to the input data and the smallest criterion value suggests the best fit we tested the new method on multivariate gaussian and real world data including mr images of normal and diseased tissue in vivo

switching strategies for sequential decision problems with multiplicative loss with application to portfolios a wide variety of problems in signal processing can be formulated such that decisions are made by sequentially taking convex combinations of vector valued observations and these convex combinations are then multiplicatively compounded over time a ldquouniversalrdquo approach to such problems might attempt to sequentially achieve the performance of the best fixed convex combination as might be achievable noncausally by observing all of the outcomes in advance by permitting different piecewise fixed strategies within contiguous regions of time the best algorithm in this broader class would be able to switch between different fixed strategies to optimize performance to the changing behavior of each individual sequence of outcomes without knowledge of the data length or the number of switches necessary the algorithms developed in this paper can achieve the performance of the best piecewise fixed strategy that can choose both the partitioning of the sequence of outcomes in time as well as the best strategy within each time segment we compete with an exponential number of such partitions using only complexity linear in the data length and demonstrate that the regret with respect to the best such algorithm is at most o ln n in the exponent where n is the data length finally we extend these results to include finite collections of candidate algorithms rather than convex combinations and further investigate the use of an arbitrary side information sequence

boussinesq system with measure forcing the paper analyzes the navier stokes system coupled with the convective diffusion equation responsible for thermal effects it is a version of the boussinesq approximation with a heat source the problem is studied in the two dimensional plane and the heat source is a measure transported by the flow for arbitrarily large initial data we prove global in time existence of unique regular solutions measure being a heat source limits regularity of constructed solutions and it requires a non standard framework of inhomogeneous besov spaces of the l infty 0 t b s p infty type the uniqueness of solutions is proved by using the lagrangian coordinates

an on line universal lossy data compression algorithm via continuous codebook refinement iii redundancy analysis for pt ii see ibid vol 42 p 822 36 1996 the gold washing data compression algorithm is an adaptive vector quantization algorithm with vector dimension n in this paper a redundancy problem of the gold washing data compression algorithm is considered it is demonstrated that for any memoryless source with finite alphabet a and generic distribution p and for any r 0 the redundancy of the gold washing data compression algorithm with dimension n defined as the difference between the average performance of the algorithm and the distortion rate function d p r of p is upper bounded by sub spl delta r sup spl delta d p r a 2 spl xi 4 log n 2n spl sigma logn n where sub spl delta r sup spl delta d p r is the partial derivative of d p r with respect to r a is the cardinality of a and spl xi 0 is a parameter used to control the threshold in the gold washing algorithm in connection with the results of zhang yang and wei see ibid vol 43 no 1 p 71 91 1997 on the redundancy of lossy source coding this shows that the gold washing algorithm has the optimal convergence rate among all adaptive finite state vector quantizers

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

the performance of universal encoding universal coding theory is surveyed from the viewpoint of the interplay between delay and redundancy the price for universality turns out to be acceptably small

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

an mdl framework for sparse coding and dictionary learning the power of sparse signal modeling with learned overcomplete dictionaries has been demonstrated in a variety of applications and fields from signal processing to statistical inference and machine learning however the statistical properties of these models such as underfitting or overfitting given sets of data are still not well characterized in the literature as a result the success of sparse modeling depends on hand tuning critical parameters for each data and application this work aims at addressing this by providing a practical and objective characterization of sparse models by means of the minimum description length mdl principle a well established information theoretic approach to model selection in statistical inference the resulting framework derives a family of efficient sparse coding and dictionary learning algorithms which by virtue of the mdl principle are completely parameter free furthermore such framework allows to incorporate additional prior information to existing models such as markovian dependencies or to define completely new problem formulations including in the matrix analysis area in a natural way these virtues will be demonstrated with parameter free algorithms for the classic image denoising and classification problems and for low rank matrix recovery in video applications however the framework is not limited to this imaging data and can be applied to a wide range of signal and data types and tasks

performance considerations for a combined information classification test using dirichlet priors the average probability of error is used to demonstrate performance of the combined bayes test cbt which assumes a uniform dirichlet prior for the symbol probabilities of each class the performance of the cbt is compared with an uncombined maximum likelihood ml based test and with the kolmogorov smirnov test

universal lossless source coding with the burrows wheeler transform the burrows wheeler transform 1994 is a reversible sequence transformation used in a variety of practical lossless source coding algorithms in each the bwt is followed by a lossless source code that attempts to exploit the natural ordering of the bwt coefficients bwt based compression schemes are widely touted as low complexity algorithms giving lossless coding rates better than those of the ziv lempel codes commonly known as lz 77 and lz 78 and almost as good as those achieved by prediction by partial matching ppm algorithms to date the coding performance claims have been made primarily on the basis of experimental results this work gives a theoretical evaluation of bwt based coding the main results of this theoretical evaluation include 1 statistical characterizations of the bwt output on both finite strings and sequences of length n spl rarr spl infin 2 a variety of very simple new techniques for bwt based lossless source coding and 3 proofs of the universality and bounds on the rates of convergence of both new and existing bwt based codes for finite memory and stationary ergodic sources the end result is a theoretical justification and validation of the experimentally derived conclusions bwt based lossless source codes achieve universal lossless coding performance that converges to the optimal coding performance more quickly than the rate of convergence observed in ziv lempel style codes and for some bwt based codes within a constant factor of the optimal rate of convergence for finite memory sources

non isothermal general ericksen leslie system derivation analysis and thermodynamic consistency we derive a model describing the evolution of a nematic liquid crystal material under the action of thermal effects the first and second laws of thermodynamics lead to an extension of the general ericksen leslie system where the leslie stress tensor and the oseen frank energy density are considered in their general forms the work postulate proposed by ericksen leslie is traduced in terms of entropy production we finally analyze the global in time well posedness of the system for small initial data in the framework of besov spaces

global well posedness to some modified 2 d nonhomogeneous navier stokes equations in this paper we are concerned with the global well posedness of modified 2d nonhomogeneous navier stokes equations our results improve the results of zhang int math res not 2008 1 rnn098 to some extent in this paper we are concerned with the global well posedness of modified 2d nonhomogeneous navier stokes equations our results improve the results of zhang int math res not 2008 1 rnn098 to some extent

universal switching linear least squares prediction in this paper we consider sequential regression of individual sequences under the square error loss we focus on the class of switching linear predictors that can segment a given individual sequence into an arbitrary number of blocks within each of which a fixed linear regressor is applied using a competitive algorithm framework we construct sequential algorithms that are competitive with the best linear regression algorithms for any segmenting of the data as well as the best partitioning of the data into any fixed number of segments where both the segmenting of the data and the linear predictors within each segment can be tuned to the underlying individual sequence the algorithms do not require knowledge of the data length or the number of piecewise linear segments used by the members of the competing class yet can achieve the performance of the best member that can choose both the partitioning of the sequence as well as the best regressor within each segment we use a transition diagram f m j willems 1996 to compete with an exponential number of algorithms in the class using complexity that is linear in the data length the regret with respect to the best member is o ln n per transition for not knowing the best transition times and o ln n for not knowing the best regressor within each segment where n is the data length we construct lower bounds on the performance of any sequential algorithm demonstrating a form of min max optimality under certain settings we also consider the case where the members are restricted to choose the best algorithm in each segment from a finite collection of candidate algorithms performance on synthetic and real data are given along with a matlab implementation of the universal switching linear predictor

on the global well posedness of 2 d inhomogeneous incompressible navier stokes system with variable viscous coefficient abstract given solenoidal vector u 0 h 2 h 1 r 2 0 1 l 2 r 2 and 0 l w 1 r r 2 with a positive lower bound for 0 1 2 and 2 r 2 1 2 we prove that 2 d incompressible inhomogeneous navier stokes system 1 1 has a unique global solution provided that the viscous coefficient 0 is close enough to 1 in the l norm compared to the size of and the norms of the initial data with smoother initial data we can prove the propagation of regularities for such solutions furthermore for 1 p 4 if 0 1 u 0 belongs to the critical besov spaces b p 1 2 p r 2 b p 1 1 2 p l 2 r 2 and the b p 1 2 p r 2 norm of 0 1 is sufficiently small compared to the exponential of u 0 l 2 2 u 0 b p 1 1 2 p we prove the global well posedness of 1 1 in the scaling invariant spaces finally for initial data in the almost critical besov spaces we prove the global well posedness of 1 1 under the assumption that the l norm of 0 1 is sufficiently small

on a lp estimate for the linearized compressible navier stokes equations with the dirichlet boundary conditions abstract in this paper a special l p estimate for the linearized compressible navier stokes in the lagrangian coordinates for the dirichlet boundary conditions is obtained the constant in the estimate does not depend on the length of time interval 0 t the result is essential to obtain an existence for regular solutions for the nonlinear problem with the lowest class of regularity in l p spaces

iterative decoding of low density parity check codes over compound channels iterative decoding of low density parity check codes over compound channels we present a setting for decoding of low density parity check ldpc codes jointly with channel estimation suitable for transmission over memoryless compound channels we show that the performance of the combined scheme can be rigorously evaluated by means of density evolution and focus on density evolution as a tool for designing a channel estimator that matches not only to the channel but also to the ldpc ensemble as well the utility of the concept is exemplified for a compound binary symmetric channel and an unknown phase additive white gaussian channel

the divergence equation in rough spaces abstract we aim at extending the existence theory for the equation div v f in a bounded or exterior domain with homogeneous dirichlet boundary conditions to a class of solutions which need not have a trace at the boundary typically the weak solutions that we shall consider will belong to some besov space b p q s with s 1 1 p 1 p after generalizing the notion of a solution for this equation we propose an explicit construction by means of the classical bogovski formula this construction enables us to keep track of a marginal information about the trace of solutions in particular it ensures that the trace is zero if f is smooth enough we expect our approach to be of interest for the study of rough solutions to systems of fluid mechanics

unique solvability for the density dependent navier stokes equations abstract in this paper we consider the incompressible navier stokes equations with a density dependent viscosity in a bounded domain of r n n 2 3 we prove the local existence of unique strong solutions for all initial data satisfying a natural compatibility condition this condition is also necessary for a very general initial data moreover we provide a blow up criterion for the regularity of the strong solution for these results the initial density need not be strictly positive it may vanish in an open subset of

global solutions for the 3 d incompressible nonhomogeneous mhd equations abstract in this article we establish the global well posedness to the 3 d incompressible non homogeneous magnetohydrodynamics mhd equations for a class of large velocity fields and magnetic fields in critical besov spaces we improve recent results by chen et al 2015 and zhai et al 2015 by extending the validity range of the parameters related to the integrability from 1 q p 6 to the seemingly optimal range 1 p q 6 we also relax their assumption on exponential norm of the initial data by replacing it by a more natural polynomial condition

on the well posedness of 3 d inhomogeneous incompressible navier stokes equations with variable viscosity abstract in this paper we mainly study the well posedness for the 3 d inhomogeneous incompressible navier stokes equations with variable viscosity with some smallness assumption on the bmo norm of the initial density we first get the local well posedness of 1 1 in the critical besov spaces moreover if the viscosity coefficient is a constant we can extend this local solution to be a global one our theorem implies that we have successfully extended the integrability index p of the initial velocity which has been obtained by abidi gui and zhang in 3 burtea in 8 and zhai and yin in 32 to approach the ideal one i e 1 p 6 the main novelty of this work is to apply the crw theorem obtained by coifman rochberg weiss in 11 to get a new a priori estimate for an elliptic equation with variable coefficients the uniqueness of the solution also relies on a lagrangian approach as in 16 17 18

speaking of infinity i i d strings we study the redundancy of three approaches to compression of independent and identically distributed i i d strings over large possibly infinite alphabets standard compression of the string itself and compression of the string s shape and pattern which describe its symbols relative magnitude and precedence respectively we determine the rate at which per symbol standard redundancy increases to infinity as the alphabet size increases show that the maximum per symbol shape redundancy is between 0 027 and 1 and compare these to results showing that per symbol pattern redundancy diminishes to zero for all alphabet sizes we relate these concepts to ordered and unordered partitions of integers and sets and use this framework to explore relations between several combinatorial quantities including the bell fubini and second type stirling numbers

space efficient representation of truncated suffix trees with applications to markov order estimation suffix trees st are useful in many text processing applications for example to determine the number of occurrences of patterns of arbitrary length in an input string x if the length n of x is large the memory required to represent the st may become a practical performance bottleneck this problem can be alleviated in cases where a nontrivial upper bound is known on the lengths of the patterns of interest by using a truncated st tst however conventional tst implementations still require n bits of memory since they store x we describe a new tst representation that avoids this limitation by storing all the information necessary to reconstruct the tst edge labels in a string y that is often much shorter than x we apply tsts to the implementation of markov order estimators where an upper bound k n on the estimated order can be derived or it is imposed for consistency for example the new representation allows for estimator implementations with sublinear space complexity in some cases of interest in other cases we show experimentally that even when the new representation does not have an asymptotic advantage it still achieves very significant memory savings in practice

global solution to the 3 d density dependent incompressible flow of liquid crystals abstract in this paper we consider the global wellposedness of the 3 d density dependent incompressible flow of liquid crystals in the critical besov spaces we first get the local solution for this system without assumptions of small density variation and then extend this local solution to be global under the smallness assumptions on initial density initial orientation field and the horizontal components of the initial velocity field this result implies the global well posedness of this system with large initial vertical velocity component

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

testing non existence of input output relationships by estimating fractal dimensions our aim is to propose tests for non existence of nonlinear relationships between signals which after passing a test can be interpreted as input and output signals of a certain system if its characteristic is sufficiently smooth the proposed tests are based on the theoretical results on equality of fractal dimensions of these signals as well as on estimation of fractal dimensions from observations they are applicable when at least one of these signals has the fractal dimensions strictly larger than one i e it is rough enough the tests are then verified on simulated data their applicability is illustrated by two sets of real data namely observations of two financial time series and samples of displacement force signals in a magneto hydrological damper

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

universal reinforcement learning we consider an agent interacting with an unmodeled environment at each time the agent makes an observation takes an action and incurs a cost its actions can influence future observations and costs the goal is to minimize the long term average cost we propose a novel algorithm known as the active lz algorithm for optimal control based on ideas from the lempel ziv scheme for universal data compression and prediction we establish that under the active lz algorithm if there exists an integer k such that the future is conditionally independent of the past given a window of k consecutive actions and observations then the average cost converges to the optimum experimental results involving the game of rock paper scissors illustrate merits of the algorithm

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

global well posedness of the two dimensional incompressible magnetohydrodynamics system with variable density and electrical conductivity abstract studied in this paper is the cauchy problem of the two dimensional magnetohydrodynamics system with inhomogeneous density and electrical conductivity it is shown that the 2 d incompressible inhomogeneous magnetohydrodynamics system with a constant viscosity is globally well posed for a generic family of the variations of the initial data and an inhomogeneous electrical conductivity moreover it is established that the system is globally well posed in the critical spaces if the electrical conductivity is homogeneous

global large solutions to 3 d inhomogeneous navier stokes system with one slow variable under a nonlinear smallness condition on the isotropic critical besov norm to the fluctuation of the initial density and the critical anisotropic besov norm of the horizontal components of the initial velocity which have to be exponentially small compared with the critical anisotropic besov norm to the third component of the initial velocity we investigate the global wellposedness of 3 d inhomogeneous incompressible navier stokes equations 1 2 in the critical besov spaces the novelty of this results is that the isotropic space structure to the inhomogeneity of the initial density function is consistent with the propagation of anisotropic regularity for the velocity field in the second part we apply the same idea to prove the global wellposedness of 1 2 with some large data which are slowly varying in one direction

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

estimation of the number of communities in the stochastic block model in this article we introduce an estimator for the number of communities in the stochastic block model sbm based on the maximization of a penalized version of the so called krichevsky trofimov mixture distribution we prove its eventual almost sure convergence to the underlying number of communities without assuming a known upper bound on that quantity our results apply to both the dense and the sparse regimes to our knowledge this is the first consistency result for the estimation of the number of communities in the sbm in the unbounded case that is when the number of communities is allowed to grow with the same size

a critical functional framework for the inhomogeneous navier stokes equations in the half space abstract this paper is devoted to solving globally the boundary value problem for the incompressible inhomogeneous navier stokes equations in the half space in the case of small data with critical regularity in dimension n 3 we state that if the initial density 0 is close to a positive constant in l w n 1 r n and the initial velocity u 0 is small with respect to the viscosity in the homogeneous besov space b n 1 0 r n then the equations have a unique global solution the proof strongly relies on new maximal regularity estimates for the stokes system in the half space in l 1 0 t b p 1 0 r n interesting for their own sake

the context tree weighting method extensions first we modify the basic binary context tree weighting method such that the past symbols x sub 1 d x sub 2 d x sub 0 are not needed by the encoder and the decoder then we describe how to make the context tree depth d infinite which results in optimal redundancy behavior for all tree sources while the number of records in the context tree is not larger than 2t 1 here t is the length of the source sequence for this extended context tree weighting algorithm we show that with probability one the compression ratio is not larger than the source entropy for source sequence length t spl rarr spl infin for stationary and ergodic sources

application of the conditional population mixture model to image segmentation the problem of image segmentation is considered in the context of a mixture of probability distributions the segments fall into classes a probability distribution is associated with each class of segment parametric families of distributions are considered a set of parameter values being associated with each class with each observation is associated an unobservable label indicating from which class the observation arose segmentation algorithms are obtained by applying a method of iterated maximum likelihood to the resulting likelihood function a numerical example is given choice of the number of classes using akaike s information criterion aic for model identification is illustrated

global unique solvability of nonhomogeneous asymmetric fluids a lagrangian approach abstract we show global existence and uniqueness of solutions for the 3d nonhomogeneous asymmetric fluids equations through a lagrangian approach in particular uniqueness of the solution is proved under quite soft assumptions about its regularity which brings the knowledge about nonhomogeneous asymmetric fluids to the same level as for the variable density navier stokes equations

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

minimax universal noiseless coding for unifilar and markov sources corresp constructive upper bounds are presented for minimax universal noiseless coding of unifilar sources without any ergodicity assumptions these bounds are obtained by quantizing the estimated probability distribution of source letters with respect to the relative entropy they apply both to fixed length to variable length fv and variable length to fixed length vf codes unifilar sources are a generalization of the usual definition of markov sources so these results apply to markov sources as well these upper bounds agree asymptotically with the lower bounds given by davisson for fv coding of stationary ergodic markov sources

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

a low complexity universal scheme for rate constrained distributed regression using a wireless sensor network we propose a scheme for rate constrained distributed nonparametric regression using a wireless sensor network the scheme is universal across a wide range of sensor noise models including unbounded and nonadditive noise it has low complexity requiring simple operations such as uniform scalar quantization with dither and message passing between neighboring nodes in the network and attains minimax optimality for regression functions in common smoothness classes we present theoretical results on the tradeoff between the compression rate communication complexity of encoding and the mse and demonstrate empirical performance of the scheme using simulations

an examination of procedures for determining the number of clusters in a data set a monte carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2 3 4 or 5 distinct nonoverlapping clusters to provide a variety of clustering solutions the data sets were analyzed by four hierarchical clustering methods external criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level thus the clustering present in the data was quite strong the simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data several procedures worked fairly well whereas others performed rather poorly thus the latter group of rules would appear to have little validity particularly for data sets containing distinct clusters applied researchers are urged to select one or more of the better criteria however users are cautioned that the performance of some of the criteria may be data dependent

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

a recursive recomputation approach for smoothing in nonlinear state space modeling an attempt for reducing space complexity in this paper we develop a new generic implementation scheme for numerical smoothing in nonlinear and bayesian state space modeling our new generic implementation scheme which we call recursive recomputation scheme reduces the space complexity from o mt to o m log t at the cost of o log t times computation of filtering distributions in time complexity this reduction is accomplished by employing carefully designed recursive recomputation the japanese stock market price time series data with t 956 is taken up as an instance to demonstrate advantage of the proposed scheme the path sampling particle smoother is implemented with the scheme to smooth the whole interval estimating the change of volatility the number of particles is 3 000 000 and the whole interval is smoothed with 5 3 gb storage accomplishing saving of storage by a factor of 1 20 the computed smoothing distribution is compared with the ones computed with the existing two other well known smoothers the forward backward smoother and the smoother based on two filter formula it turns out that among the three ours is the only method which succeeded in computing a reliable and plausible smoothing distribution in the situation

stability of nontrivial solutions of the navier stokes system on the three dimensional torus abstract this paper examines the stability of nontrivial regular solutions to the navier stokes equations on a three dimensional torus it is shown that the w2 1r norm of the perturbation can be controlled if its initial data are small enough in the l2 norm a key element of the proof is to apply the marcinkiewicz theorem to obtain the estimate for the stokes system in particular we prove the stability of unforced two dimensional flows

on the redundancy of lossy source coding with abstract alphabets the redundancy problem of lossy source coding with abstract source and reproduction alphabets is considered for coding at a fixed rate level it is shown that for any fixed rate r 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the rate r such that the distortion redundancy of c sub n defined as the difference between the performance of c sub n and the distortion rate function d p r of p is upper bounded by spl part d p r spl part r ln n 2n o ln n n for coding at a fixed distortion level it is demonstrated that for any d 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the fixed distortion d such that the rate redundancy of c sub n defined as the difference between the performance of c sub n and the rate distortion function r p d of p is upper bounded by 7ln n 6n o ln n n these results strengthen the traditional berger s 1968 1971 abstract alphabet source coding theorem and extend the positive redundancy results of zhang yang and wei see ibid vol 43 no 1 p 71 91 1997 and ibid vol 42 p 803 21 1996 on lossy source coding with finite alphabets and the redundancy result of wyner see ibid vol 43 p 1452 64 1997 on block coding of memoryless gaussian sources

global well posedness for the 3d incompressible inhomogeneous navier stokes equations and mhd equations abstract the present paper is dedicated to the global well posedness for the 3d inhomogeneous incompressible navier stokes equations in critical besov spaces without smallness assumption on the variation of the density we aim at extending the work by abidi gui and zhang 2012 2 and 2013 3 to a lower regularity index about the initial velocity the key to that improvement is a new a priori estimate for an elliptic equation with nonconstant coefficients in besov spaces which have the same degree as l 2 in r 3 finally we also generalize our well posedness result to the inhomogeneous incompressible mhd equations

an on line universal lossy data compression algorithm via continuous codebook refinement i basic results a new on line universal lossy data compression algorithm is presented for finite memoryless sources with unknown statistics its performance asymptotically approaches the fundamental rate distortion limit the codebook is generated on the fly and continuously adapted by simple rules there is no separate codebook training or codebook transmission candidate codewords are randomly generated according to an arbitrary and possibly suboptimal distribution through a carefully designed gold washing or information theoretic sieve mechanism good codewords and only good codewords are promoted to permanent status with high probability we also determine the rate at which our algorithm approaches the fundamental limit

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

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

noise histogram and cluster validity for gaussian mixtured data abstract in this study a critique of the clustering methodology is carried out for the definition of a cluster determination of the number of clusters and evaluation of heuristic partitional clustering algorithms when the data is a noisy gaussian mixture the effects of noise in determining the number of clusters and the clustering parameters are investigated two cluster validity criteria namely the likelihood information criterion and the sum of squared error are described it is concluded that these criteria can be used as a guide in deciding on the number of valid clusters by using the proposed sum of squared error criterion an improvement algorithm which reduces the effect of noise on the results of heuristic clustering algorithms is described

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

coding for a binary independent piecewise identically distributed source two weighting procedures are presented for compaction of output sequences generated by binary independent sources whose unknown parameter may occasionally change the resulting codes need no knowledge of the sequence length t i e they are strongly sequential and also the number of parameter changes is unrestricted the additional transition redundancy of the first method was shown to achieve the merhav lower bound i e log t bits per transition for the second method we could prove that additional transition redundancy is not more than 3 2 log t bits per transition which is more than the merhav bound however the storage and computational complexity of this method are also more interesting than those of the first method simulations show that the difference in redundancy performance between the two methods is negligible

global well posedness of 3 d inhomogeneous navier stokes system with initial velocity being a small perturbation of 2 d solenoidal vector field abstract motivated by 21 we consider the global wellposedness to the 3 d incompressible inhomogeneous navier stokes equations with large horizontal velocity in particular we proved that when the initial density is close enough to a positive constant then given divergence free initial velocity field of the type v 0 h 0 x h w 0 h w 0 3 x h x 3 we shall prove the global wellposedness of 1 1 the main difficulty here lies in the fact that we will have to obtain the l 1 r lip r 3 estimate for convection velocity in the transport equation of 1 1 toward this and due to the strong anisotropic properties of the approximate solutions we will have to work in the framework of anisotropic littlewood paley theory here

global well posedness to the 3 d incompressible inhomogeneous navier stokes equations with a class of large velocity in this article we consider the global well posedness to the 3 d incompressible inhomogeneous navier stokes equations with a class of large velocity more precisely assuming a0 bq 13q r3 and u0 u0h u03 bp 1 1 3p r3 for p q 1 6 with sup 1p 1q 13 inf 1p 1q we prove that if ca0bq 13q u03bp 1 1 3p 1 1 c u0hbp 1 1 3p u03bp 1 1 3p1 u0hbp 1 1 3p 1 then the system has a unique global solution a c 0 bq 13q r3 u c 0 bp 1 1 3p r3 l1 r bp 11 3p r3 it improves the recent result of m paicu and p zhang j funct anal 262 3556 3584 2012 where the exponent form of the initial smallness condition is replaced by a polynomial form

global solution to the incompressible oldroyd b model in the critical l p framework the case of the non small coupling parameter this paper is dedicated to the global well posedness issue of the incompressible oldroyd b model in the whole space mathbb r d with d geqq 2 it is shown that this set of equations admits a unique global solution in a certain critical lp type besov space provided that the initial data but not necessarily the coupling parameter is small enough as a consequence even through the coupling effect between the equations of velocity u and the symmetric tensor of constrains is not small one may construct the unique global solution to the oldroyd b model for a class of large highly oscillating initial velocity the proof relies on the estimates of the linearized systems of u and u mathbb p rm div tau which may be of interest for future works this result extends the work by chemin and masmoudi siam j math anal 33 84 112 2001 to the non small coupling parameter case

regularity results for two dimensional flows of multiphase viscous fluids global regularity results for weak solutions of the navier stokes equations for two dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive lower bounds on the initial density as an application we deduce regularity properties for the integral curves of the corresponding velocity field finally we prove regularity results in the small for strong solutions

universal portfolios with side information we present a sequential investment algorithm the spl mu weighted universal portfolio with side information which achieves to first order in the exponent the same wealth as the best side information dependent investment strategy the best state constant rebalanced portfolio determined in hindsight from observed market and side information outcomes this is an individual sequence result which shows the difference between the exponential growth wealth of the best state constant rebalanced portfolio and the universal portfolio with side information is uniformly less than d 2n log n 1 k n log 2 for every stock market and side information sequence and for all time n here d k m 1 is the number of degrees of freedom in the state constant rebalanced portfolio with k states of side information and m stocks the proof of this result establishes a close connection between universal investment and universal data compression

a vector quantization approach to universal noiseless coding and quantization a two stage code is a block code in which each block of data is coded in two stages the first stage codes the identity of a block code among a collection of codes and the second stage codes the data using the identified code the collection of codes may be noiseless codes fixed rate quantizers or variable rate quantizers we take a vector quantization approach to two stage coding in which the first stage code can be regarded as a vector quantizer that quantizes the input data of length n to one of a fixed collection of block codes we apply the generalized lloyd algorithm to the first stage quantizer using induced measures of rate and distortion to design locally optimal two stage codes on a source of medical images two stage variable rate vector quantizers designed in this way outperform standard one stage fixed rate vector quantizers by over 9 db the tail of the operational distortion rate function of the first stage quantizer determines the optimal rate of convergence of the redundancy of a universal sequence of two stage codes we show that there exist two stage universal noiseless codes fixed rate quantizers and variable rate quantizers whose per letter rate and distortion redundancies converge to zero as k 2 n sup 1 log n when the universe of sources has finite dimension k this extends the achievability part of rissanen s theorem from universal noiseless codes to universal quantizers further we show that the redundancies converge as o n sup 1 when the universe of sources is countable and as o n sup 1 spl epsiv when the universe of sources is infinite dimensional under appropriate conditions

on the redundancy of the fixed database lempel ziv algorithm for spl phi mixing sources the redundancy problem of the fixed database lempel ziv 1977 algorithm is considered it is demonstrated that for a class of spl phi mixing sources which includes markov sources unifilar sources and finite state sources as special cases the redundancy of the fixed database lempel ziv algorithm with database size n is lower bounded by h loglogn logn o logloglogn logn and upper bounded by 2h loglogn logn o logloglogn logn where h is the entropy rate of the source the method of proof is new and uses the concept of variable length to variable length codes

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