interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations we present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three a suitable weak solution is regular near an interior point z if the scaled lx tp q norm of the velocity with 1 3 p 2 q 2 1 q is sufficiently small near z and if the scaled lx tl m norm of the magnetic field with 1 3 l 2 m 2 1 m is bounded near z similar results are also obtained for the vorticity and for the gradient of the vorticity furthermore with the aid of the regularity criteria we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations

a second order perceptron algorithm kernel based linear threshold algorithms such as support vector machines and perceptron like algorithms are among the best available techniques for solving pattern classification problems in this paper we describe an extension of the classical perceptron algorithm called second order perceptron and analyze its performance within the mistake bound model of on line learning the bound achieved by our algorithm depends on the sensitivity to second order data information and is the best known mistake bound for efficient kernel based linear threshold classifiers to date this mistake bound which strictly generalizes the well known perceptron bound is expressed in terms of the eigenvalues of the empirical data correlation matrix and depends on a parameter controlling the sensitivity of the algorithm to the distribution of these eigenvalues since the optimal setting of this parameter is not known a priori we also analyze two variants of the second order perceptron algorithm one that adaptively sets the value of the parameter in terms of the number of mistakes made so far and one that is parameterless based on pseudoinverses

remarks on the singular set of suitable weak solutions for the three dimensional navier stokes equations abstract in this study s denotes the possible interior singular set of suitable weak solutions for the three dimensional navier stokes equations we improve the known upper box counting dimension of this set from 360 277 1 300 given by 24 to 975 758 1 286 we also show that s r log e r 0 0 27 113 which extends the previous corresponding results concerning the improvement of the classical caffarelli kohn nirenberg theorem by a logarithmic factor given by choe and lewis 3 j funct anal 175 348 369 2000 and by choe and yang 4 comm math phys 336 171 198 2015 the proof is inspired by a new e regularity criterion which was proved by guevara and phuc 7 calc var 56 68 2017

on the construction of suitable solutions to the navier stokes equations and questions regarding the definition of large eddy simulation abstract this paper discusses the notion of large eddy simulation les as a mathematical concept and proposes to use the concept of suitable weak solutions as a building block for the development of a mathematical theory of les various techniques for constructing suitable weak solutions to the three dimensional navier stokes equations are reviewed and many features of these mathematical constructions are shown to comply with well accepted heuristic characterizations of les

multipliers paramultipliers and weak strong uniqueness for the navier stokes equations abstract in this article we describe spaces p such that if u is a weak in the sense of leray j leray sur le mouvement d un fluide visqueux remplissant l espace acta math 63 1934 193 248 solution of the navier stokes system for some initial data u 0 and if u belongs to p then u is unique in the class of weak solutions we say then that weak strong uniqueness holds it turns out that the proof of such results relies on the boundedness of a trilinear functional f l 2 h l 2 h p r where belong to 0 1 in order to find optimal conditions for the boundedness of f we are led to describing spaces of multipliers and of paramultipliers that is functions which map by classical pointwise product or by paraproduct a given sobolev spaces in another given sobolev space the study of these spaces enables us to give conditions for weak strong uniqueness which generalise all previously known results from the famous serrin criterion j serrin the initial value problem for the navier stokes equations in r e langer ed nonlinear problems univ of wisconsin press 1963 pp 69 98 to the recent conditions formulated by lemarie rieusset p g lemarie rieusset recent developments in the navier stokes problem chapman and hall 2003

hausdorff measure of boundary singular points in the magnetohydrodynamic equations abstract we study boundary singular points of suitable weak solutions to the three dimensional incompressible magnetohydrodynamic equations by using the generalised hausdorff measure we estimate the size of boundary singular points and present the improved range of powers of logarithmic factors

weak solution to the steady compressible flow of nematic liquid crystals abstract in this study three dimensional equations for the steady compressible flow of nematic liquid crystals are considered in a bounded domain for the adiabatic exponent 1 we establish the existence of a weak solution by three level approximation and adapting the weak convergence method

extension criterion via two components of vorticity on strong solutions to the 3 d navier stokes equations we shall show that only two components of vorticity play an essential role to determine possibility of extension of the time interval for the local strong solution to the navier stokes equations then we shall apply our extension theorem to regularity criterion on weak solutions due to serrin and beirao da veiga chae choe proved the same criterion as beirao da veiga only by means of the two components of vorticity we deal with the critical case which they excluded our criterion may be regarded as the generalization of the result of beal kato majda from l to bmo

non uniform elliptic equations in convex lipschitz domains abstract non uniform elliptic equations in convex lipschitz domains are concerned the non smooth domains consist of a periodic connected high permeability sub region and a periodic disconnected matrix block subset with low permeability let 0 1 denote the size ratio of the matrix blocks to the whole domain and let 2 0 1 denote the permeability ratio of the disconnected matrix block subset to the connected sub region the w 1 p norm for p 1 of the elliptic solutions in the high permeability sub region is shown to be bounded uniformly in however the w 1 p norm of the solutions in the low permeability subset may not be bounded uniformly in roughly speaking if the sources in the low permeability subset are small enough the solutions in that subset are bounded uniformly in otherwise the solutions cannot be bounded uniformly in relations between the sources and the variation of the solutions in the low permeability subset are also presented in this work

local regularity criteria of the 3d navier stokes and related equations abstract in this paper we are concerned with the local regularity of suitable weak solutions to the 3d navier stokes equations and its generalized case with fractional power of the laplacian 3 4 1 in the first part we derive some regularity criteria in terms of the deformation tensor d u for the classical navier stokes equations in the second part for the fractional case we obtain some regularity conditions for suitable weak solutions including the velocity u the gradient of the velocity u the rotation tensor u and the deformation tensor d u this generalizes the results obtained by gustafson et al 2007

generalized mhd equations abstract solutions of the d dimensional generalized mhd gmhd equations t u u u p b b u t b u b b u b are studied in this paper we pay special attention to the impact of the parameters and on the regularity of solutions our investigation is divided into three major cases 1 0 and 0 2 0 and 0 and 3 0 and 0 when 0 and 0 the gmhd equations with any 0 and 0 possess a global weak solution corresponding to any l2 initial data furthermore weak solutions associated with 1 2 d 4 and 1 2 d 4 are actually global classical solutions when their initial data are sufficiently smooth as a special consequence smooth solutions of the 3d gmhd equations with 5 4 and 5 4 do not develop finite time singularities the study of the gmhd equations with 0 and 0 is motivated by their potential applications in magnetic reconnection a local existence result of classical solutions and several global regularity conditions are established for this case these conditions are imposed on either the vorticity u or the current density j b but not both and are weaker than some of current existing ones when 0 and 0 the gmhd equations reduce to the ideal mhd equations it is shown here that the ideal mhd equations admit a unique local solution when the prescribed initial data is in a holder space cr with r 1

on the singular set in the navier stokes equations we consider suitably weak solutions u p to the incompressible navier stokes equations and under various assumptions on u obtain estimates for the size of its singular set one of our results improves a well known theorem of caffarelli kohn and nirenberg

a new approximate maximal margin classification algorithm a new incremental learning algorithm is described which approximates the maximal margin hyperplane w r t norm p 2 for a set of linearly separable data our algorithm called alma p approximate large margin algorithm w r t norm p takes o p 1 2 2 corrections to separate the data with p norm margin larger than 1 where g is the normalized p norm margin of the data alma p avoids quadratic or higher order programming methods it is very easy to implement and is as fast as on line algorithms such as rosenblatt s perceptron algorithm we performed extensive experiments on both real world and artificial datasets we compared alma 2 i e alma p with p 2 to standard support vector machines svm and to two incremental algorithms the perceptron algorithm and li and long s romma the accuracy levels achieved by alma 2 are superior to those achieved by the perceptron algorithm and romma but slightly inferior to svm s on the other hand alma 2 is quite faster and easier to implement than standard svm training algorithms when learning sparse target vectors alma p with p 2 largely outperforms perceptron like algorithms such as alma 2

a regularity criterion at one scale without pressure for suitable weak solutions to the navier stokes equations abstract in this paper we continue our work in 15 to derive e regularity criteria at one scale without pressure for suitable weak solutions to the navier stokes equations we establish an e regularity criterion below of suitable weak solutions for any 0 q 1 u 5 2 d x d t e as an application we extend the previous corresponding results concerning the improvement of the classical caffarelli kohn nirenberg theorem by a logarithmic factor

exponentiated gradient versus gradient descent for linear predictors we consider two algorithm for on line prediction based on a linear model the algorithms are the well known gradient descent gd algorithm and a new algorithm which we call eg they both maintain a weight vector using simple updates for the gd algorithm the update is based on subtracting the gradient of the squared error made on a prediction the eg algorithm uses the components of the gradient in the exponents of factors that are used in updating the weight vector multiplicatively we present worst case loss bounds for eg and compare them to previously known bounds for the gd algorithm the bounds suggest that the losses of the algorithms are in general incomparable but eg has a much smaller loss if only a few components of the input are relevant for the predictions we have performed experiments which show that our worst case upper bounds are quite tight already on simple artificial data

hausdorff measure of the singular set in the incompressible magnetohydrodynamic equations we derive a local energy inequality for weak solutions of the three dimensional magnetohydrodynamic equations combining biot savart law interpolation inequalities and the local energy inequality we prove a partial regularity theorem for suitable weak solutions furthermore we obtain an improved estimate for the logarithmic hausdorff dimension of the singular set of suitable weak solutions

existence and regularity of a class of weak solutions to the navier stokes equations abstract we construct a class of weak solutions to the navier stokes equations which have second order spatial derivatives and one order time derivatives of p power summability for 1 p 5 4 meanwhile we show that u l s 0 t w 2 r with 1 s 3 2 r 2 for 1 r 5 4 r can be relaxed not to exceed 3 2 if we consider only in the interior of in the end we extend the classical regularity theorem our results show that u is a regular solution if u l s 0 t l r with 1 s 3 2 r 1 for satisfying 1 3 with 1 s 1 r 5 6 for arbitrary domain in r 3 and 1 s 2 for r n with n 3 this result was previously obtained byh beirao da veiga chinese ann math ser b 16 1995 407 412

on the regularity of weak solutions to the magnetohydrodynamic equations abstract in this paper we study the regularity of weak solution to the incompressible magnetohydrodynamic equations we obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations which is similar to that of incompressible navier stokes equations moreover our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto hydrodynamic equations

on a larger class of stable solutions to the navier stokes equations in exterior domains where w w x w1 x wn x and x denote the unknown velocity vector and the unknown pressure of the fluid at point x respectively while f f x f i j x i j 1 n is the given tensor with div f n j 1 f 1 j xj nj 1 f n j xj denoting the external force in our previous paper 23 we established the theory of existence uniqueness and regularity for solutions to n s0 in the class w ln

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

uniform estimates in two periodic homogenization problems we analyze two partial differential equations that are posed on perforated domains we provide a priori estimates that do not depend on the size of the perforation a sequence of solutions is uniformly bounded in a sobolev space of regular functions the first homogenization problem concerns the laplace and the mean curvature operator with neumann boundary conditions we derive uniform lipschitz estimates for the solutions the result is used in the analysis of a free boundary system of fluid mechanics a contractive iteration yields the existence of solutions and uniform estimates the key is the use of function spaces that are different from the usual lp spaces 2000 john wiley sons inc

two regularity criteria for the 3d mhd equations this work establishes two regularity criteria for the 3d incompressible mhd equations the first one is in terms of the derivative of the velocity field in one direction while the second one requires suitable boundedness of the derivative of the pressure in one direction

the perceptron algorithm versus winnow linear versus logarithmic mistake bounds when few input variables are relevant abstract we give an adversary strategy that forces the perceptron algorithm to make kn mistakes in learning monotone disjunctions over n variables with at most k literals in contrast littlestone s algorithm winnow makes at most o k log n mistakes for the same problem both algorithms use thresholded linear functions as their hypotheses however winnow does multiplicative updates to its weight vector instead of the additive updates of the perceptron algorithm in general we call an algorithm additive if its weight vector is always a sum of a fixed initial weight vector and some linear combination of already seen instances thus the perceptron algorithm is an example of an additive algorithm we show that an adversary can force any additive algorithm to make n k 1 2 mistakes in learning a monotone disjunction of at most k literals simple experiments show that for k n winnow clearly outperforms the perceptron algorithm also on nonadversarial random data

global regularity to the 3d mhd equations with large initial data in bounded domains this paper considers the global regularity to the 3d incompressible mhd equations with large initial data in bounded domains let u and b denote the viscosity coefficient magnetic diffusivity velocity field and magnetic field respectively we construct new systems for u b and u b to overcome the difficulties caused by the large initial data it is shown that u b h1 is globally bounded as long as u0 b0 h1 1 or u0 b0 h1 1 is sufficiently small which indicates that the navier stokes equations can be regularized by the magnetic field

global weak solutions and attractors of the three dimensional maxwell bloch two level laser systems the three dimensional maxwell bloch system governs the multi longitudinal and transverse mode dynamics of two level wide aperture lasers in an optical ring cavity the system is hyperbolic in the propagation direction and dispersive in the transverse directions due to diffraction effects a rich variety of optical patterns and chaos are present in the dynamics we show the global existence of weak solutions inlp 2 p spaces of the maxwell bloch system under both absorbing and periodic boundary conditions the weak solutions are unique within the class of solutions provided by our regularization procedure and approach a universal attractor which has only partial smoothing instead of thec smoothing property found in early works for the longitudinal one dimensional and transverse two dimensional cases the idea of the proof makes essential use of both the hyperbolicity and dispersivity of the system in the case of periodic boundary condition our result depends on a conjectural strichartz inequality

partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations in this paper we study the local behavior of the solutions to the three dimensional magnetohydrodynamic equations we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions it is shown that in some neighborhood of x0 t0 the gradients of the velocity field u and the magnetic field b are locally uniformly bounded in l norm as long as that either the scaled local l2 norm of the gradient or the scaled local total energy of the velocity field is small and the scaled local total energy of the magnetic field is uniformly bounded these estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory as an immediately corollary we can derive an estimates of hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid various partial regularity results are obtained as consequences of our blow up estimates

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

remarks on the regularity criterion to the navier stokes equations via the gradient of one velocity component abstract in this paper we consider the classical three dimensional 3d incompressible navier stokes equations by making use of a new anisotropic sobolev inequality we improve a regularity criterion for the weak solution to the 3d navier stokes equations involving the gradient of one velocity component this result improves the previous work of zhou and pokorn 2010

on the number of singular points of weak solutions to the navier stokes equations we consider a suitable weak solution to the three dimensional navier stokes equations in the space time cylinder 0 t let be the set of singular points for this solution and t x t for a given open subset and for a given moment of time t 0 t we obtain an upper bound for the number of points of the set t 2001 john wiley sons inc

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

existence of suitable weak solutions of complex ginzburg landau equations and properties of the set of singular points in this paper we consider the supercritical complex ginzburg landau equation we discuss the existence of suitable weak solution in where is a bounded domain in rn or the whole space we also discuss the properties of the set of the singular points of the suitable weak solution in rn which means that the possible singular points are located in a bounded ball for any given time and there is no singular point on the whole space after limited time

existence of weak and regular solutions for keller segel system with degradation coupled to fluid equations abstract we establish the global well posedness for the following chemotaxis fluid system t n u n n n c n q t c u c c c n t u u u p u n u 0 in r d d 2 3 where 0 q 2 1 d and 0 1 for either q 2 d 1 2 or q 2 d 0 3 we prove the global existence of regular solutions in case that q 2 1 d and 0 very weak solutions are constructed as well

tracking the best linear predictor in most on line learning research the total on line loss of the algorithm is compared to the total loss of the best off line predictor u from a comparison class of predictors we call such bounds static bounds the interesting feature of these bounds is that they hold for an arbitrary sequence of examples recently some work has been done where the predictor ut at each trial t is allowed to change with time and the total on line loss of the algorithm is compared to the sum of the losses of ut at each trial plus the total cost for shifting to successive predictors this is to model situations in which the examples change over time and different predictors from the comparison class are best for different segments of the sequence of examples we call such bounds shifting bounds they hold for arbitrary sequences of examples and arbitrary sequences of predictors naturally shifting bounds are much harder to prove the only known bounds are for the case when the comparison class consists of a sequences of experts or boolean disjunctions in this paper we develop the methodology for lifting known static bounds to the shifting case in particular we obtain bounds when the comparison class consists of linear neurons linear combinations of experts our essential technique is to project the hypothesis of the static algorithm at the end of each trial into a suitably chosen convex region this keeps the hypothesis of the algorithm well behaved and the static bounds can be converted to shifting bounds

convergence results for a class of spectrally hyperviscous models of 3 d turbulent flow abstract we consider the spectrally hyperviscous navier stokes equations shnse which add hyperviscosity to the nse but only to the higher frequencies past a cutoff wavenumber m 0 in guermond and prudhomme 2003 18 subsequence convergence of shnse galerkin solutions to dissipative solutions of the nse was achieved in a specific spectral vanishing viscosity setting our goal is to obtain similar results in a more general setting and to obtain convergence to the stronger class of leray solutions in particular we obtain subsequence convergence of shnse strong solutions to leray solutions of the nse by fixing the hyperviscosity coefficient while the spectral hyperviscosity cutoff m 0 goes to infinity this formulation presents new technical challenges and we discuss how its motivation can be derived from computational experiments e g those in borue and orszag 1996 1998 3 4 we also obtain weak subsequence convergence to leray weak solutions under the general assumption that the hyperviscous coefficient goes to zero with no constraints imposed on the spectral cutoff in both of our main results the aubin compactness theorem provides the underlying framework for the convergence to leray solutions

regularity and singularity in solutions of the three dimensional navier stokes equations higher moments of the vorticity field m t in the form of l 2 m norms graphic 1 are used to explore the regularity problem for solutions of the three dimensional incompressible navier stokes equations on the domain graphic 2 it is found that the set of quantities formula 3 provide a natural scaling in the problem resulting in a bounded set of time averages d m t on a finite interval of time 0 t the behaviour of d m 1 d m is studied on what are called good and bad intervals of 0 t which are interspersed with junction points neutral i for large but finite values of m with large initial data m 0 o gr 4 it is found that there is an upper bound formula 4 which is punctured by infinitesimal gaps or windows in the vertical walls between the good bad intervals through which solutions may escape while this result is consistent with that of leray leray 1934 acta math 63 193 248 doi 10 1007 bf02547354 5 and scheffer scheffer 1976 pacific j math 66 535 552 this estimate for m corresponds to a length scale well below the validity of the navier stokes equations 1 embed inline graphic 1 gif 2 embed inline graphic 2 gif 3 embed graphic 1 gif 4 embed graphic 2 gif 5 http dx doi org doi 10 1007 bf02547354

some new a priori estimates for second order elliptic and parabolic interface problems abstract we present some new a priori estimates of the solutions to the second order elliptic and parabolic interface problems the novelty of these estimates lies in the explicit appearance of the discontinuous coefficients and the jumps of coefficients across the interface

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

regularity criteria for suitable weak solutions of the navier stokes equations near the boundary abstract we present some new regularity criteria for suitable weak solutions of the navier stokes equations near the boundary in dimension three we prove that suitable weak solutions are holder continuous up to the boundary provided that the scaled mixed norm l x t p q with 3 p 2 q 2 2 q p q 3 2 is small near the boundary our methods yield new results in the interior case as well partial regularity of weak solutions is also analyzed under some additional integral conditions

gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients in this paper we derive global w1 and piecewise c1 estimates for solutions to divergence form elliptic equations with piecewise holder continuous coefficients the novelty of these estimates is that even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients they are independent of the distance between these surfaces

interior regularity criteria for suitable weak solutions of the navier stokes equations we present new interior regularity criteria for suitable weak solutions of the 3 d navier stokes equations a suitable weak solution is regular near an interior point z if either the scaled l p q x t norm of the velocity with 3 p 2 q 2 1 q or the l p q x t norm of the vorticity with 3 p 2 q 3 1 q or the l p q x t norm of the gradient of the vorticity with 3 p 2 q 4 1 q 1 p is sufficiently small near z

forward self similar solutions of the fractional navier stokes equations abstract we study forward self similar solutions to the 3 d navier stokes equations with the fractional diffusion first we construct a global time forward self similar solutions to the fractional navier stokes equations with 5 6 1 for arbitrarily large self similar initial data by making use of the so called blow up argument moreover we prove that this solution is smooth in r 3 0 in particular when 1 we prove that the solution constructed by korobkov and tsai 2016 16 satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system which implies this solution has the same properties as a solution which was constructed in jia and sverak 2014 13

on the existence of lp solutions of the magnetohydrodynamic equations in a bounded domain abstract in this paper we show the existence of lp solutions of the magnetohydrodynamic equations with the non slip boundary condition and the perfect conducting wall boundary condition the magnetohydrodynamic equations are a phenomenological model for magnetic fluid

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

on the weak strong uniqueness of koch tataruʼs solution for the navier stokes equations abstract we prove the weak strong uniqueness between koch tataru s solution and leray s weak solution for three dimensional incompressible navier stokes equations

on the l2 decay of weak solutions for the 3d asymmetric fluids equations abstract we study the long time behavior of weak solutions for the asymmetric fluids equations in the whole space r 3 we prove that u w t l 2 r 3 2 c t 1 3 2 for all t 0 through fourier splitting moreover we use a direct method to show how to take advantage of the particular form of the equation for the angular velocity and obtain the so far unknown faster decay rate w t l 2 r 3 2 c t 1 5 2 for all t 0

the p norm generalization of the lms algorithm for adaptive filtering recently much work has been done analyzing online machine learning algorithms in a worst case setting where no probabilistic assumptions are made about the data this is analogous to the h sup spl infin setting used in adaptive linear filtering bregman divergences have become a standard tool for analyzing online machine learning algorithms using these divergences we motivate a generalization of the least mean squared lms algorithm the loss bounds for these so called p norm algorithms involve other norms than the standard 2 norm the bounds can be significantly better if a large proportion of the input variables are irrelevant i e if the weight vector we are trying to learn is sparse we also prove results for nonstationary targets we only know how to apply kernel methods to the standard lms algorithm i e p 2 however even in the general p norm case we can handle generalized linear models where the output of the system is a linear function combined with a nonlinear transfer function e g the logistic sigmoid

regularity criterion of weak solutions to the 3d navier stokes equations via two velocity components abstract this paper is concerned with the regularity criterion of leray hopf weak solutions to the 3d navier stokes equations with respect to serrin type condition on two velocity filed components it is shown that the weak solution u u 1 u 2 u 3 is regular on 0 t if there exist two solution components for example u 2 and u 3 satisfying the condition u 2 u 3 l p 1 0 t l q 1 r 3 for 2 p 1 3 q 1 2 3 2 q 1

regularity criteria of the magnetohydrodynamic equations in bounded domains or a half space abstract we study three dimensional incompressible magnetohydrodynamic equations in bounded domains or a half space we present new regularity criteria of weak solutions a pair of weak solutions u b become regular if u satisfies serrin s type conditions when we consider no slip or slip boundary conditions for the velocity field u and slip boundary conditions for the magnetic field b in either bounded domains or a half space in addition in the case of a half space with no slip boundary conditions for u and slip boundary conditions for b we demonstrate that if tangential components of u and normal component of b satisfy serrin s type conditions then a pair of weak solutions u b become regular

elliptic equations with bmo coefficients in lipschitz domains in this paper we study inhomogeneous dirichlet problems for elliptic equations in divergence form optimal regularity requirements on the coefficients and domains for the w 1 p 1 p estimates are obtained the principal coefficients are supposed to be in the john nirenberg space with small bmo semi norms the domain is supposed to have lipschitz boundary with small lipschitz constant these conditions for the w 1 p theory do not just weaken the requirements on the coefficients they also lead to a more general geometric condition on the domain

on partial regularity results for the navier stokes equations looking at the regularity results of scheffer respectively caffarelli kohn and nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the navier stokes equations

intermittency and regularity issues in 3d navier stokes turbulence two related open problems in the theory of 3d navier stokes turbulence are discussed in this paper the first is the phenomenon of intermittency in the dissipation field dissipation range intermittency was first discovered experimentally by batchelor and townsend over fifty years ago it is characterized by spatio temporal binary behaviour in which long quiescent periods in the velocity signal are interrupted by short active events during which there are violent fluctuations away from the average the second and related problem is whether solutions of the 3d navier stokes equations develop finite time singularities during these events this paper shows that leray s weak solutions of the three dimensional incompressible navier stokes equations can have a binary character in time the time axis is split into good and bad intervals on the good intervals solutions are bounded and regular whereas singularities are still possible within the bad intervals an estimate for the width of the latter is very small and decreases with increasing reynolds number it also decreases relative to the lengths of the good intervals as the reynolds number increases within these bad intervals lower bounds on the local energy dissipation rate and other quantities such as u t and u t are very large resulting in strong dynamics at sub kolmogorov scales intersections of bad intervals for n 1 are related to the potentially singular set in time it is also proved that the navier stokes equations are conditionally regular provided in a given bad interval the energy has a lower bound that is decaying exponentially in time

long time behavior for the nonstationary navier stokes flows in l1 r n the large time decay for the navier stokes flows in l1 r n is a long standing unsolved question the main difficulties are that usual lq lr estimates for the stokes flow fail in this case and the projection operator p l1 r n l 1 r n becomes unbounded using the stokes solution formula we find a crucial and new estimate for the stokes flow in l1 r n which plays a fundamental role in studying the time l1 behavior for the navier stokes equations in addition we decompose the operator p into two parts and reduce its unboundedness to establish an l1 estimate for an elliptic problem with neumann boundary condition which is overcome by using the weighted estimates of the gaussian kernel s convolution the main results in this article are motivated by bae s works 2006 2008 1 2 and brandolese s work 2001 11

linear elliptic equations in composite media with anisotropic fibres abstract linear elliptic equations in composite media with anisotropic fibres are concerned the media consist of a periodic set of anisotropic fibres with low conductivity included in a connected matrix with high conductivity inside the anisotropic fibres the conductivity in the longitudinal direction is relatively high compared with that in the transverse directions the coefficients of the elliptic equations depend on the conductivity this work is to derive the holder and the gradient l p estimates uniformly in the period size of the set of anisotropic fibres as well as in the conductivity ratio of the fibres in the transverse directions to the connected matrix for the solutions of the elliptic equations furthermore it is shown that inside the fibres the solutions have higher regularity along the fibres than in the transverse directions

existence and decay of solutions to the dissipative quasi geostrophic equation with delays abstract we consider the dissipative quasi geostrophic equation with forcing and advective terms containing delays we prove the existence of weak solutions and algebraic decay of the energy provided the initial data 0 is in l p l 2 1 p 2 and the forcing term satisfies some mild conditions on its long time behavior

boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations abstract we present some new regularity criteria for suitable weak solutions of magnetohydrodynamic equations near boundary in dimension three we prove that suitable weak solutions are holder continuous near boundary provided that either the scaled l x t p q norm of the velocity with 3 p 2 q 2 2 q or the scaled l x t p q norm of the vorticity with 3 p 2 q 3 2 q are sufficiently small near the boundary

pressure representation and boundary regularity of the navier stokes equations with slip boundary condition abstract we first represent the pressure in terms of the velocity in r 3 using this representation we prove that a solution to the navier stokes equations is in l r 3 0 under the critical assumption that u l loc r r 3 r 2 r 1 with r 3 while for r 3 the smallness is required in h j choe boundary regularity of weak solutions of the navier stokes equations j differential equations 149 2 1998 211 247 a boundary l estimate for the solution is derived if the pressure on the boundary is bounded in our work we remove the boundedness assumption of the pressure here our estimate is local indeed employing moser type iteration and the reverse holder inequality we find an integral estimate for l norm of u

on regularity criteria of the navier stokes equations in bounded domains we present new regularity criteria of weak solutions to the navier stokes equations in bounded domains of rn n 3 4 we prove that if a weak solution u u1 u2 un satisfies uk lx tpk qk with 2 qk n pk 1 n pk for k 1 2 n 1 and un xn lx tl m with 2 m n l 2 n 2 l n then u becomes smooth

minimal initial data for potential navier stokes singularities abstract assuming some initial data u 0 h 1 2 r 3 lead to a singularity for the 3d navier stokes equations we show that there are also initial data with the minimal h 1 2 norm which will produce a singularity

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

unbounded normal derivative for the stokes system near boundary we study local boundary regularity for the stokes system we show that unlike in the interior case non local effects can lead to a violation of local regularity in the spatial variables near the boundary

efficient learning with virtual threshold gates abstract we reduce learning simple geometric concept classes to learning disjunctions over exponentially many variables we then apply an online algorithm called winnow whose number of prediction mistakes grows only logarithmically with the number of variables the hypotheses of winnow are linear threshold functions with one weight per variable we find ways to keep the exponentially many weights of winnow implicitly so that the time for the algorithm to compute a prediction and update its virtual weights is polynomial our method can be used to learnd dimensional axis parallel boxes whendis variable and unions ofd dimensional axis parallel boxes whendis constant the worst case number of mistakes of our algorithms for the above classes is optimal to within a constant factor and our algorithms inherit the noise robustness of winnow we think that other online algorithms with multiplicative weight updates whose loss bounds grow logarithmically with the dimension are amenable to our methods