homogenization of two phase fluid flow in porous media via volume averaging abstract a technique of local volume averaging is employed to obtain general equations which depict mass and momentum transport of incompressible two phase flow in porous media starting from coupled navier stokes cahn hilliard equations for incompressible two phase fluid flow the averaging is performed without oversimplifying either the porous media or the fluid mechanical relations the resulting equations are darcy s law for two phase flow with medium parameters which could be evaluated by experiment the richards equation of the mixed form can be deduced from the resulting equations the differences between the resulting equations and the empirical two phase fluid flow model adopted in oil industry are discussed by several numerical examples

generalized entropic uncertainty relations a new class of uncertainty relations is derived for pairs of observables in a finite dimensional hilbert space which do not have any common eigenvector this class contains an entropic uncertainty relation which improves a previous result of deutsch and confirms a recent conjecture by kraus some comments are made on the extension of these relations to the case where the hilbert space is infinite dimensional

laws of large numbers for semimartingales with applications to stochastic regression strong laws of large numbers for matrix normalised vector valued local martingales are established the results are derived from strong laws for positive local submartingales and purely discontinuous local martingales and a borel cantelli type lemma for local martingales of finite variation the multivariate strong laws are applied to study strong consistency of estimates in stochastic linear regression models

a law of large numbers for identically distributed martingale differences the averages of an identically distributed martingale difference sequence converge in mean to zero but the almost sure convergence of the averages characterizes l log l in the following sense if the terms of an identically distributed martingale difference sequence are in l log l the averages converge to zero almost surely but if f is any integrable random variable with zero expectation which is not in l log l there is a martingale difference sequence whose terms have the same distributions as f and whose averages diverge almost surely the maximal function of the averages of an identically distributed martingale difference sequence is integrable if its terms are in l log l the converse is false

a two parameter maximal ergodic theorem with dependence let x 1 x 2 ldots and y 1 y 2 ldots be independent sequences of i i d u 0 1 random variables we characterize completely those borel functions f on lbrack 0 1 rbrack 2 for which the strong law of large numbers and the maximal ergodic theorem hold for the doubly indexed family 1 nm sum i leq n j leq m f x i y j

two parameter strong laws and maximal inequalities for u statistics we provide necessary and sufficient conditions for two parameter convergence in the strong law of large numbers for u statistics we also obtain weak type 1 1 inequalities for one and two sample u statistics of order 2 which are in a sense best possible

conditional generalizations of strong laws which conclude the partial sums converge almost surely suppose that for every independent sequence of random variables satis fying some hypothesis condition h it follows that the partial sums converge almost surely then it is shown that for every arbitrarily dependent sequence of random variables the partial sums converge almost surely on the event where the conditional distributions given the past satisfy precisely the same condition h thus many strong laws for independent sequences may be immediately generalized into conditional results for arbitrarily dependent sequences 1 introduction if every sequence of independent random variables having property a has property b almost surely does every arbitrarily dependent sequence of random variables have property b almost surely on the set where the conditional distributions have property a not in general but comparisons of the conditional borel cantelli lemmas the condi tional three series theorem and many martingale results with their independent counter parts suggest that the answer is affirmative in fairly general situations the purpose of this note is to prove theorem 1 which states in part that if property b is the partial sums converge then the answer is always affirmative regardless of property a thus many strong laws for independent sequences even laws yet undiscovered may be immediately generalized into conditional results for arbitrarily dependent sequences

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

entropic heisenberg limits and uncertainty relations from the holevo information bound strong and general entropic and geometric heisenberg limits are obtained for estimates of multiparameter unitary displacements in quantum metrology such as the estimation of a magnetic field from the induced rotation of a probe state in three dimensions a key ingredient is the holevo bound on the shannon mutual information of a quantum communication channel this leads to a bayesian bound on performance in terms of the prior distribution of the displacement and the asymmetry of the input probe state with respect to the displacement group a geometric measure of performance related to entropy is proposed for general parameter estimation it is also shown how strong entropic uncertainty relations for mutually unbiased observables such as number and phase position and momentum energy and time and orthogonal spin 1 2 directions can be obtained from elementary applications of holevo s bound a geometric interpretation of results is emphasised in terms of the volumes of quantum and classical statistical ensembles

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

invariance principles for von mises and u statistics the almost sure approximation of von mises statistics and u statistics by appropriate stochastic integrals with respect to kiefer processes is obtained in general these integrals are non gaussian processes as applications we get almost sure versions for the estimator of the variance and for the 2 test of goodness of fit

the capacity of the quantum channel with general signal states it is shown that the capacity of a classical quantum channel with arbitrary possibly mixed states equals the maximum of the entropy bound with respect to all a priori distributions this completes the recent result of hausladen jozsa schumacher westmoreland and wootters 1996 who proved the equality for the pure state channel

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

an efficient time adaptivity based on chemical potential for surface cahn hilliard equation using finite element approximation abstract we present numerical simulations for the surface cahn hilliard equation which describes phase separation phenomenon occurred on general surfaces the spatial discretization is based on surface finite element method while the temporal discretization methods are first and second order stabilized semi implicit schemes which guarantee the free energy decay an efficient and parameter free adaptive time stepping strategy is proposed based on the numerical energy stability generated by stabilized semi implicit scheme the main idea is to use discrete chemical potential the byproduct of stabilized semi implicit scheme to estimate the variation of numerical energy that is used as a indicator to update the time step the operation avoid calculating numerical energy with gauss integral in each time step and reduce the calculated cost in addition optimal error estimate of first order stabilized semi implicit scheme in the case of curved surface are provided finally numerical experiments are presented to demonstrate the stability accuracy and efficiency of the proposed algorithms

uniform regularity of the weak solution to higher order navier stokes cahn hilliard systems abstract higher order navier stokes cahn hilliard system is a noted interface system that describes the evolution of two immiscible incompressible fluids regularity of the solutions to this system is a primary issue from a mathematical point of view the purpose of this paper is to improve the known results regarding the regularity with respect to time of weak solutions to two dimensional higher order navier stokes cahn hilliard system namely h 1 h k regularity of the first and second derivatives of the weak solutions to two dimensional higher order navier stokes cahn hilliard system is obtained with the methods of energy estimates

error estimates with smooth and nonsmooth data for a finite element method for the cahn hilliard equation a finite element method for the cahn hilliard equation a semilinear parabolic equation of fourth order is analyzed both in a spatially semidiscrete case and in a completely discrete case based on the backward euler method error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data a detailed study of the regularity of the exact solution is included the analysis is based on local lipschitz conditions for the nonlinearity with respect to sobolev norms and the existence of a ljapunov functional for the exact and the discretized equations is essential a result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup upper semicontinuity with respect to the discretization parameters is obtained as a simple application of the nonsmooth data error estimate

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

extended quantum conditional entropy and quantum uncertainty inequalities quantum states can be subjected to classical measurements whose incompatibility or uncertainty can be quantified by a comparison of certain entropies there is a long history of such entropy inequalities between position and momentum recently these inequalities have been generalized to the tensor product of several hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized our proofs utilize the technique of the original derivation of strong subadditivity of the von neumann entropy

entropic energy time uncertainty relation energy time uncertainty plays an important role in quantum foundations and technologies and it was even discussed by the founders of quantum mechanics however standard approaches e g robertson s uncertainty relation do not apply to energy time uncertainty because in general there is no hermitian operator associated with time following previous approaches we quantify time uncertainty by how well one can read off the time from a quantum clock we then use entropy to quantify the information theoretic distinguishability of the various time states of the clock our main result is an entropic energy time uncertainty relation for general time independent hamiltonians stated for both the discrete time and continuous time cases our uncertainty relation is strong in the sense that it allows for a quantum memory to help reduce the uncertainty and this formulation leads us to reinterpret it as a bound on the relative entropy of asymmetry because of the operational relevance of entropy we anticipate that our uncertainty relation will have information processing applications

fully discrete finite element approximations of the navier stokes cahn hilliard diffuse interface model for two phase fluid flows this paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the navier stokes equations and the cahn hilliard equation which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids in the model the two sets of equations are coupled through an extra phase induced stress term in the navier stokes equations and a fluid induced transport term in the cahn hilliard equation fully discrete mixed finite element methods are proposed for approximating the coupled system it is shown that the proposed numerical methods satisfy a mass conservation law and a discrete energy law which is analogous to the basic energy law for the phase field model the convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law as a by product the convergence result also provides a constructive proof of the existence of weak solutions to the navier stokes cahn hilliard phase field model numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

long time numerical simulations for phase field problems using p adaptive spectral deferred correction methods a high order and energy stable scheme is developed to simulate phase field models by combining the semi implicit spectral deferred correction sdc method and the energy stable convex splitting technique the convex splitting scheme we use here is a linear unconditionally stable method but is only of first order accuracy so the sdc method can be used to iteratively improve the rate of convergence however it is found that the accuracy improvement may affect the overall energy stability which is intrinsic to the phase field models to compromise the accuracy and stability a local p adaptive strategy is proposed to enhance the accuracy by sacrificing some local energy stability in an acceptable level the proposed strategy is found very useful for producing accurate numerical solutions at small time dynamics as well as long time steady state with reasonably large time stepsizes numerical experiments are carried out to demonstrate the high effectiveness of the proposed numerical strategy

glivenko cantelli properties of some generalized empirical df s and strong convergence of generalized l statistics we study a nonclassical form of empirical df hnwhich is of u statistic structure and extend to hnthe classical exponential probability inequalities and glivenko cantelli convergence properties known for the usual empirical df an important class of statistics is given byt hn where t is a generalized form of l functional for such statisticswe prove almost sure convergence using an approach which separates the functional analytic and stochastic components of the problem and handles the latter component by application of glivenko cantelli type properties classical results for u statistics and l statistics are obtained as special cases without addition of unnecessary restrictions many important new types of statistics of current interest are covered as well by our result

uniform h2 regularity of solution for the 2d navier stokes cahn hilliard phase field model abstract in this paper we provide the uniform h 2 regularity results with respect to small perturbation parameter e of the solution and its time derivative for the 2d navier stokes cahn hilliard phase field model with t 0 based on sharp a priori estimates for the solution of model under weak assumptions on the initial values we show that the h 2 regularity of the solution and its time derivative depends on e 1 but is independent of time t

decoupling of banach valued multilinear forms in independent symmetric banach valued random variables let e be a banach space and e be symmetric continuous and convex let ui and ri be independent sequences of random variables having respectively u 0 1 and symmetric bernoulli distributions and let u i j and r i j for j 1 2 d be independent copies of these sequences we prove two sided inequalities between the quantities e phi sum limits underset raise0 3em hbox smash scriptscriptstyle i in mathbb z d r i 1 r i d f underset raise0 3em hbox smash scriptscriptstyle i u i 1 u i d and their decoupled versions e phi sum limits underset raise0 3em hbox smash scriptscriptstyle iota in mathbb z d r i 1 1 r i d d f underset raise0 3em hbox smash scriptscriptstyle iota u i 1 1 u i d d for bochner integrable f i 0 1 d e this generalizes results of kwapien and of zinn

convergence and stability of finite element nonlinear galerkin method for the navier stokes equations in this article we present a new fully discrete finite element nonlinear galerkin method which are well suited to the long time integration of the navier stokes equations spatial discretization is based on two grid finite element technique time discretization is based on euler explicit scheme with variable time step size moreover we analyse the boundedness convergence and stability condition of the finite element nonlinear galerkin method our discussion shows that the time step constraints of the method depend only on the coarse grid parameter h and the time step constraints of the finite element galerkin method depend on the fine grid parameter h h under the same convergence accuracy

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

analysis of finite element approximations of a phase field model for two phase fluids this paper studies a phase field model for the mixture of two immiscible and incompressible fluids the model is described by a nonlinear parabolic system consisting of the nonstationary stokes equations coupled with the allen cahn equation through an extra phase induced stress term in the stokes equations and a fluid induced transport term in the allen cahn equation both semi discrete and fully discrete finite element methods are developed for approximating the parabolic system it is shown that the proposed numerical methods satisfy a discrete energy law which mimics the basic energy law for the phase field model error estimates are derived for the semi discrete method and the convergence to the phase field model and to its sharp interface limiting model are established for the fully discrete finite element method by making use of the discrete energy law numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach

necessary and sufficient conditions for the strong law of large numbers for u statistics under some mild regularity on the normalizing sequence we obtain necessary and sufficient conditions for the strong law of large numbers for symmetrized u statistics we also obtain necessary and sufficient conditions for the a s convergence of series of an analogous form

analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits we propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science the primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method in particular by focusing on the dependence of the error bounds on the parameter e known as the measure of the interface thickness optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size h and the time step size k in particular it is shown that all error bounds depend on 1 a only in some lower polynomial order for small e the cruxes of the analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of chen and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect finally as a nontrivial byproduct the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients the sharp interface limits include the classical stefan problem the generalized stefan problems with surface tension and surface kinetics the motion by mean curvature flow and the hele shaw model

error analysis of a mixed finite element method for the cahn hilliard equation we propose and analyze a semi discrete and a fully discrete mixed finite element method for the cahn hilliard equation ut u 1f u 0 where 0 is a small parameter error estimates which are quasi optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain in particular it is shown that all error bounds depend on only in some lower polynomial order for small the cruxes of our analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of alikakos and fusco 2 and chen 15 to prove a discrete counterpart of it for a linearized cahn hilliard operator to handle the nonlinear term on a stretched time grid the ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the hele shaw mullins sekerka problem as 0 in 29