products of commuting boolean algebras of projections and banach space geometry new criteria and banach spaces are presented for example gl spaces and banach spaces with property alpha that ensure that the boolean algebra generated by a pair of bounded commuting boolean algebras of projections is itself bounded the notion of r boundedness plays a fundamental role it is shown that the strong operator closure of any r bounded boolean algebra of projections is necessarily bade complete also for a dedekind sigma complete banach lattice e the boolean algebra consisting of all band projections in e is r bounded if and only if e has finite cotype in this situation every bounded boolean algebra of projections in e is r bounded and has a bade complete strong closure

the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

martingale transforms and complex uniform convexity les transformees de martingale et les operateurs integraux singuliers de calderon zygmund sont bornees comme des operateurs de l 2 l 1 dans l 2 l q quand 0 q 1 si y est un sous espace reflexif de l 1 alors l 1 y peut se renormer pour etre 2 complexe uniformement convexe on donne une nouvelle demonstration de la propriete de cotype 2 de l 1 h

solving differential equations by a maximum entropy minimum norm method with applications to fokker planck equations the method of maximum entropy minimum norm is utilized to produce a general method of solving differential equations the technique is a generalization and extension of previous work performed by baker jarvis j math phys 30 302 1989 it is found that introducing an additional constraint on the norm of the solution vector produces a probability distribution that is integrable over the entire real axis a number of simplifications occur in this extended method the lagrange multipliers and solution vector can be solved for explicitly thus eliminating the necessity of solving systems of nonlinear equations for the lagrange multipliers as was required in the previous approach it is shown that the solution obtained is equivalent to a minimum norm approximation the maximum entropy solution of differential equations with fourier moments is shown to be identical to a fourier series solution additionally the new method is applied to solving the random walk and fokker planck equations

the swift hohenberg equation on conic manifolds abstract we consider the swift hohenberg equation on manifolds with conical singularities and show existence uniqueness and maximal regularity of the short time solution in terms of mellin sobolev spaces moreover we give a necessary and sufficient condition so that the above solution exists for all times space asymptotic expansion of the solution near the singularity is also provided and its relation to the local geometry is shown the same problem is considered on closed manifolds and similar results are obtained by using the above singular analysis theory

the role of information measures in the determination of the maximum entropy minimum norm solution of the generalized inverse problem we discuss here the role of different information measures vis a vis the maximum entropy minimum norm method of baker jarvis et al bj for dealing with under determined inverse linear transforms it is shown that the structure of the approximate solutions provided by the bj approach does not depend upon the functional form of the information measure

pseudodifferential operators on bochner spaces and an application we consider pseudodifferential operators with operator valued symbols a x acting on a umd banach space x assuming some regularity of holder type in x and mihlin type in we prove l p n x boundedness of such operators this result is then applied to the study of l p maximal regularity for nonautonomous parabolic evolution equations

new thoughts on old results of r t seeley in this paper we show that the lp realization of a vector valued elliptic boundary value problem j admits a bounded calculus on lp g e 1 p provided the top order coefficients of are holder continuous here g denotes a domain in n with compact c2m boundary and e a banach space of class our proof is based on an abstract perturbation result for operators admitting bounded calculus and kernel estimates for the solution of j

maximum entropy principle and classical evolution equations with source terms abstract we devise a maximum entropy technique to construct approximate time dependent solutions to evolution equations endowed with source terms and consequently not preserving normalization in some special cases the method yields exact solutions it is shown that the present implementation of the maximum entropy prescription always even in the case of approximate solutions preserves the exact functional relationship between the time derivative of the entropy and the time dependent solutions of the evolution equation other properties of the maximum entropy solutions and some illustrative examples are also discussed

maximum entropy approach to the collisional vlasov equation exact solutions we show that the maximum entropy approach proposed by el wakil elhanbaly and abdou from now on eea in s a el wakil a elhanbaly m a abdou physica a 323 2003 213 for solving approximately the collisional vlasov equation actually can provide exact solutions if properly implemented we consider here two alternative procedures for obtaining exact maximum entropy solutions of the aforementioned equation on the one hand after identifying an appropriate set of relevant mean values moments we show that there are exact maximum entropy solutions associated with that set of moments these solutions can be studied focusing either on the equations of motion of the moments themselves or on the equations of motion of the corresponding lagrange multipliers on the other hand it is possible to find exact solutions of the reduced equation considered by eea if one takes explicitly into account the zeroth order moment of the solutions

an improved lattice boltzmann method for simulating advective diffusive processes in fluids abstract lattice boltzmann methods are widely used to simulate advective diffusive processes in fluids lattice bhatnagar gross krook methods presented in the literature mostly just exhibit first order spatial accuracy and introduce errors proportional to the velocity squared formulations proposed to alleviate this have only been partly successful and are valid only in certain specific situations we present and demonstrate here a formulation that produces no such second order errors this formulation suggests that a subtle but important adjustment is all it takes to improve the accuracy of the method the key to the improved accuracy of this new model is the non standard definition of the concentration that relates to the distribution function describing the advection diffusion in lattice boltzmann the main advantage of the algorithm comes to view when simulating situations where fluid density variations appear the present formulation of the advection diffusion algorithm will by taking into account these fluid density variations drastically reduce the errors produced compared to the standard formulations we also show how a source term is included in this new formulation without it losing its second order spatial accuracy

properties of q entropies basic properties of the q entropy sq q 1 1 1 tr q 0 q 1 for states of a quantum system are established concavity quasi convexity continuity and failure of additivity and subadditivity for composite systems

non standard thermal statistics with q entropies we consider the quantum thermal statistics a la gibbs shannon szilard jaynes based on q entropies sq q 1 1 1 tr q 0 q 1 and the non standard internal energy functionals uq tr qh proposed by c tsallis j stat phys 52 479 487 1988

discrete lattice effects on the forcing term in the lattice boltzmann method we show that discrete lattice effects must be considered in the introduction of a force into the lattice boltzmann equation a representation of the forcing term is then proposed with the representation the navier stokes equation is derived from the lattice boltzmann equation through the chapman enskog expansion several other existing force treatments are also examined

a new scheme for source term in lbgk model for convection diffusion equation a new scheme for source term in the lattice bgk lbgk model for convection diffusion equation is proposed unlike the models proposed previously the present scheme only requires the source term in order of the knudsen number by adding a differential operator of the source term to the evolution equation the scheme can be applied to reaction diffusion systems directly numerical results are found to be in excellent agreement with the analytical solutions it is also found that the numerical accuracy of the present scheme is generally much better than that of the existing models

consistent second order boundary implementations for convection diffusion lattice boltzmann method in this study an alternative second order boundary scheme is proposed under the framework of the convection diffusion lattice boltzmann lb method for both straight and curved geometries with the proposed scheme boundary implementations are developed for the dirichlet neumann and linear robin conditions in a consistent way the chapman enskog analysis and the hermite polynomial expansion technique are first applied to derive the explicit expression for the general distribution function with second order accuracy then the macroscopic variables involved in the expression for the distribution function is determined by the prescribed macroscopic constraints and the known distribution functions after streaming see the paragraph after eq 29 for the discussions of the streaming step in lb method after that the unknown distribution functions are obtained from the derived macroscopic information at the boundary nodes for straight boundaries boundary nodes are directly placed at the physical boundary surface and the present scheme is applied directly when extending the present scheme to curved geometries a local curvilinear coordinate system and first order taylor expansion are introduced to relate the macroscopic variables at the boundary nodes to the physical constraints at the curved boundary surface in essence the unknown distribution functions at the boundary node are derived from the known distribution functions at the same node in accordance with the macroscopic boundary conditions at the surface therefore the advantages of the present boundary implementations are i the locality i e no information from neighboring fluid nodes is required ii the consistency i e the physical boundary constraints are directly applied when determining the macroscopic variables at the boundary nodes thus the three kinds of conditions are realized in a consistent way it should be noted that the present focus is on two dimensional cases and theoretical derivations as well as the numerical validations are performed in the framework of the two dimensional five velocity lattice model

functional calculus for semigroup generators via transference abstract in this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators in particular it is proved that if a generates a c 0 semigroup on a hilbert space then for each 0 the operator a has a bounded calculus for the closed ideal of bounded holomorphic functions on a sufficiently large right half plane that satisfy f z o e re z as z the bound of this calculus grows at most logarithmically as 0 as a consequence f a is a bounded operator for each holomorphic function f on a right half plane with polynomial decay at then we show that each semigroup generator has a so called strong m bounded calculus for all m n and that this property characterizes semigroup generators similar results are obtained if the underlying banach space is a umd space upon restriction to so called bounded semigroups the hilbert space results actually hold in general banach spaces

invariant of dynamical systems a generalized entropy in this work the concept of entropy of a dynamical system as given by kolmogorov is generalized in the sense of tsallis it is shown that this entropy is an isomorphism invariant being complete for bernoulli schemes

sharp interface limit for a stokes allen cahn system we consider the sharp interface limit of a coupled stokes allen cahn system when a parameter epsilon 0 that is proportional to the thickness of the diffuse interface tends to zero in a two dimensional bounded domain for sufficiently small times we prove convergence of the solutions of the stokes allen cahn system to solutions of a sharp interface model where the interface evolution is given by the mean curvature equation with an additional convection term coupled to a two phase stokes system with an additional contribution to the stress tensor which describes the capillary stress to this end we construct a suitable approximation of the solution of the stokes allen cahn system using three levels of the terms in the formally matched asymptotic calculations and estimate the difference with the aid of a suitable refinement of a spectral estimate of the linearized allen cahn operator moreover a careful treatment of the coupling terms is needed

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

nonequilibrium scheme for computing the flux of the convection diffusion equation in the framework of the lattice boltzmann method in this paper we propose a local nonequilibrium scheme for computing the flux of the convection diffusion equation with a source term in the framework of the multiple relaxation time mrt lattice boltzmann method lbm both the chapman enskog analysis and the numerical results show that at the diffusive scaling the present nonequilibrium scheme has a second order convergence rate in space a comparison between the nonequilibriumschemeandtheconventionalsecond ordercentral differenceschemeindicatesthat althoughboth schemes have a second order convergence rate in space the present nonequilibrium scheme is more accurate than thecentral differencescheme inaddition thefluxcomputationrenderedbythepresentschemealsopreservesthe parallel computation feature of the lbm making the scheme more efficient than conventional finite difference schemes in the study of large scale problems finally a comparison between the single relaxation time model and the mrt model is also conducted and the results show that the mrt model is more accurate than the single relaxation time model both in solving the convection diffusion equation and in computing the flux

solution to boundary value problems using the method of maximum entropy the method of maximum entropy is used to solve a class of linear boundary value problems the method is based on using various moments of the differential equation as constraints when maximizing the entropy various examples are presented and compared to exact solutions for varying numbers of moments it is found that the maximum entropy approximation is in many cases better than a fourier series solution for a given number of expansion terms and moments the method is very general and will find applications in many areas of physics a comparison of the amount of work necessary for the maximum entropy solution versus finite difference techniques is presented and it is found that the maximum entropy technique shows promise as an alternative solution technique

initial and boundary conditions for the lattice boltzmann method an alternative approach of implementing initial and boundary conditions for the lattice boltzmann method is presented the basic idea is to calculate the lattice boltzmann populations at a boundary node from the fluid variables that are specified at this node using the gradients of the fluid velocity the numerical performance of the lattice boltzmann method is tested on several problems with exact solutions and is also compared to an explicit finite difference projection method the discretization error of the lattice boltzmann method decreases quadratically with finer resolution both in space and in time the roundoff error of the lattice boltzmann method creates problems unless double precision arithmetic is used

on measuring unboundedness of the h calculus for generators of analytic semigroups we investigate the boundedness of the h infty calculus by estimating the bound b e of the mapping h infty b x f f a t e for e near zero here a generates the analytic semigroup t and h infty is the space of bounded analytic functions on a domain strictly containing the spectrum of a we show that b e o loge in general whereas b e o 1 for bounded calculi this generalizes a result by vitse and complements work by haase and rozendaal for non analytic semigroups we discuss the sharpness of our bounds and show that single square function estimates yield b e o loge

a solution to the problem of l p maximal regularity we give a negative solution to the problem of the l p maximal regularity on various classes of banach spaces including l q spaces with 1 q neq 2 infty

the lattice boltzmann equation a new tool for computational fluid dynamics abstract we present a series of applications which demonstrate that the lattice boltzmann equation is an adequate computational tool to address problems spanning a wide spectrum of fluid regimes ranging from laminar to fully turbulent flows in two and three dimensions

lattice boltzmann method for diffusion limited partial dissolution of fluids a lattice boltzmann model for two partially miscible fluids is developed by partially miscible we mean that although there is a definite interfacial region separating the two fluids with a surface tension force acting at all points of the transition region each fluid can nonetheless accept molecules from the other fluid up to a set solubility limit we allow each fluid to diffuse into the other with the solubility and diffusivity in each fluid being input parameters the approach is to define two regions within the fluid one interfacial region having finite width across which most of the concentration change occurs and in which a surface tension force and color separation step are allowed for and one miscible fluid region where the concentration of the binary fluids follows an advection diffusion equation and the mixture as a whole obeys the navier stokes incompressible flow equations numerical examples are presented in which the algorithm produces results that are quantitatively compared to exact analytical results as well as qualitatively examined for their reasonableness the model has the ability to simulate how bubbles of one fluid flow through another while dissolving their contents as well as to simulate a range of practical invasion problems such as injecting supercritical co 2 into a porous material saturated with water for sequestration purposes

the operator valued marcinkiewicz multiplier theorem and maximal regularity given a closed linear operator on a umd space we characterize maximal regularity of the non homogeneous problem

numerical study of lattice boltzmann methods for a convection diffusion equation coupled with navier stokes equations numerous lattice boltzmann lb methods have been proposed for solution of the convection diffusion equations cde for the 2d problem d2q9 d2q5 or d2q4 velocity models are usually used when lb convection diffusion models are used to solve a cde coupled with navier stokes equations boundary conditions are found to be critically important for accurately solving the coupled simulations following the idea of a regularized scheme latt et al 2008 phys rev e 77 056703 a regularized boundary condition for solving a cde is proposed a simple extrapolation scheme is also proposed for the neumann boundary condition spatial accuracies of three existing and the proposed boundary conditions are discussed in details the numerical evaluations are based on simulations of steady and unsteady natural convection flows in a cavity and an unsteady taylor couette flow our studies show that the simplest d2q4 model with terms of o u in the equilibrium distribution function is capable of obtaining results of equal accuracy as d2q5 or d2q9 models for the cde a slightly revised lb equation for solving a cde that is used to cancel some unwanted terms does not seem to be necessary for incompressible flows the regularized boundary condition for solving the cde has second order spatial accuracy and it is the best one in terms of the spatial accuracy the regularized scheme and non equilibrium extrapolation scheme are applicable to handle both the dirichlet and neumann boundary conditions for the neumann boundary condition with zero flux all the five boundary conditions are applicable to give accurate results and the bounce back scheme is the simplest one

on the closedness of the sum of two closed operators o t t 1 where e g a is the infinitesimal generator of an analytic semigroup in a complex banach space x we find that in general it is false that for every fe lp 0 t x problem 1 has a solution uewi p o t x c lp o t a usually it is necessary to impose some further regularity conditions on f consider for instance the way in which problem 1 is studied in the paper 8 and compare th 3 7 and 3 14 which are applied in w 4 of the same paper to solve problem 1 we find that in general we have only strong solutions while in the hilbert space setting we have strict solutions this corresponds to the fact that in general the operator a b where b u u is only closable while in a hilbert space it is closed this remark suggests that the better result may depend on the nicer geometric properties of a hilbert space with respect to a general banach space and it is natural to ask which conditions should be imposed on the banach space to force the sum of two suitable closed opera tors to be closed the above considerations motivate the manifold aim of the present paper a to find conditions on a banach space x and on closed linear operators a and b acting in x which ensure that a b is closed b to apply the result of a to problem 1 it seems that the convenient property for the banach space x is convexity or otherwise stated the umd condition for information about these equiva lent conditions we refer to 4 in which it is also proved that if x is convex and l p oe then the hilbert transform is continuous from lp r x into lp r x on the other hand it is also known see 2 that if there exists pe 1 oe such that the hilbert transform is continuous from lp r x into itself

statistical treatment of autonomous systems with divergencelless flows abstract statistical mechanics sm has been quite successful in providing one with exact or at least approximate descriptions of the time dependent solutions of the liouville or of the von neumann equation within the context of hamiltonian dynamics hd here we investigate how to extend some of its ideas to more general dynamical systems we find that for systems that retain from hd just the divergenceless character of the phase space flow many ideas borrowed from the maximum entropy principle approach to sm apply in such a generalized context as well

a reaction flow lattice boltzmann model the authors describe a development of a lattice boltzmann fluid flow model enabling the simulation of the reaction and flow of low concentration species in two dimensions in principle any number of species and reactions can be implemented in this model in making this development they derive an expression for the solution of the diffusion equation using the lattice boltzmann approach

generalized statistical mechanics connection with thermodynamics the authors show the manner through which the recent generalization by tsallis 1988 of the boltzmann gibbs statistics becomes consistent with a generalized thermodynamics preserving the legendre transformation framework of standard thermodynamics in addition to that they generalize the shannon additivity

bounded h infty calculus for a class of nonlocal operators the bidomain operator in the l_q setting the bidomain operator mathbb a a nonlocal operator is studied in a bounded domain omega subset mathbb r d with boundary partial omega of class c 2 within the l q setting for 1 q infty assuming a fairly general framework it is shown that this operator is sectorial invertible on functions with mean zero and admits a bounded h infty calculus with h infty angle 0 in particular it has the property of maximal l p l q regularity and the fractional power domains mathcal d mathbb a alpha are identified furthermore it is shown that mathbb a generates a strongly continuous compact analytic and exponentially stable semigroup on l q 0 omega with q independent purely countable point spectrum sigma mathbb a