lower bound for the online bin packing problem with restricted repacking in 1996 ivkovic and lloyd a fundamental restriction on fully dynamic maintenance of bin packing inform process lett 59 1996 pp 229 232 gave the lower bound frac 4 3 on the asymptotic worst case ratio for so called fully dynamic bin packing algorithms where the number of repackable items in each step is restricted by a constant in this paper we improve this result to about 1 3871 we present our proof for a semionline case of the classical bin packing but it works for fully dynamic bin packing as well we prove the lower bound by analyzing and solving a specific optimization problem the bound can be expressed exactly using the lambert w function

tensor train versus monte carlo for the multicomponent smoluchowski coagulation equation in this paper we present a novel numerical algorithm for the space homogeneous multicomponent multidimensional smoluchowski coagulation equation the number of components is considered as dimensionality the new methodology is based on the classical finite difference predictor corrector scheme in a straightforward implementation of this scheme however one would have to compute and store prohibitively many values of the grid function at the nodes of a multidimensional grid we propose to use special low parametric representations for the grid functions and as well for the coagulation kernel the corresponding multidimensional arrays are approximated by low rank tensor train decompositions reducing them to combinations of small low dimensional arrays eventually to matrices for which we can use fast algorithms of linear algebra instead of o n 2 d operations in the classical scheme we propose a new method that requires only o d 2 n log n operations where n is the number of nodes per axis in the space grid and d is the number of components in this work we accelerate the predictor corrector time scheme and use the trapezoidal rule for the computation of multidimensional integral operators thus the accuracy of the new method is o h 2 2 where h is the space grid step and is the time step

two dimensional online bin packing with rotation in two dimensional bin packing problems the input items are rectangles which need to be packed in a non overlapping manner the goal is to assign the items into unit squares using an axis parallel packing most previous work on online packing concentrated on items of fixed orientation which must be assigned such that their bottom side is parallel to the bottom of the bin in this paper we study the case of rotatable items which can be rotated by ninety degrees we give almost tight bounds on the asymptotic competitive ratio of bounded space bin packing of rotatable items and introduce a new unbounded space algorithm this improves the results of fujita and hada

an asymptotic competitive scheme for online bin packing in the online bin packing problem a list of items with integral sizes between 1 to b arrive one by one each item must be irrevocably assigned into a bin of capacity b upon its arrival without any information on the subsequent items and the goal is to minimize the number of used bins in this paper we deal with online bin packing from a new sight we present an asymptotic competitive scheme i e for any 0 the asymptotic competitive ratio is at most where is the smallest possible asymptotic competitive ratio among all online algorithms apart from the technical results the analysis to bridge the online and the off line approaches might be of particular interests

on a class of continuous coagulation fragmentation equations abstract a model for the dynamics of a system of particles undergoing simultaneously coalescence and breakup is considered each particle being assumed to be fully identified by its size existence of solutions to the corresponding evolution integral partial differential equation is shown for product type coagulation kernels with a weak fragmentation the failure of density conservation or gelation is also investigated in some particular cases

information theoretic analysis of iterated bayesian acoustic source localization in a static ocean waveguide fundamental constructs of information theory are applied to quantify the performance of iterated sequential bayesian localization of a time harmonic source in a range and time invariant acoustic waveguide using the segmented fourier transforms of the received pressure time series the nonlinear relation defined by acoustic propagation between the source location and the received narrowband spectral components is treated as a nonlinear communication channel the performance analysis includes mismatch between the acoustic channel and the model channel on which the bayesian inference is based source location uncertainty is quantified by the posterior probability density of source location by the posterior entropy and associated uncertainty area by the information gain relative entropy at each iteration and by large ensemble limits of these quantities a computational example for a vertical receiver array in a shallow water waveguide is presented with acoustic propagation represented by normal mode

numerical solution of the smoluchowski kinetic equation and asymptotics of the distribution function we obtain the numerical solution of the smoluchowski kinetic equation for model kernels u m1 m2 varies as m1 m2 lambda and u m1 m2 varies as m1m2 lambda 2 0 lambda or 2 we show that the behaviour of the solution for the kernel u varies as m1 m2 lambda at 0 lambda or 1 and u varies as m1m2 lambda 2 at 0 lambda or 2 becomes self similar after some time the shape of the scaling function is analysed in particular a simple approximate expression for it at u varies as m1 m2 lambda is found an interesting result is obtained for u varies as m1m2 lambda 2 0 lambda 1 the asymptotic behaviour of the scaling function proved to be non power we develop the procedure for determining tcr the critical indices and the exponent of the power law asymptotics of the smoluchowski equation solution the concrete values of these quantities for the model kernels are obtained the stability conditions of the algorithm for numerical solving are analysed the possibilities afforded by numerical solution for investigation of the smoluchowski equation are discussed

numerical simulation of the smoluchowski coagulation equation in this paper we develop a numerical scheme for the smoluchowski coagulation equation which relies on a conservative formulation and a finite volume approach several numerical simulations are performed to test the validity of the scheme and the expected behavior of the model in particular the gelation phenomenon and the long time behavior of the solution are numerically studied

algorithms for the relaxed online bin packing model the typical online bin packing problem requires the fitting of a sequence of rationals in 0 1 into a minimum number of bins of unit capacity by packing the ith input element without any knowledge of the sizes or the number of input elements that follow moreover unlike typical online problems this one issue does not admit any data reorganization i e no element can be moved from one bin to another in this paper first of all the relaxed online bin packing model will be formalized this model allows a constant number of elements to move from one bin to another as a consequence of the arrival of a new input element then in the context of this new model two online algorithms will be described the first presents linear time and space complexities with a 1 5 approximation ratio and moves at most once only small elements the second instead is an o n log n time and linear space algorithm with a 1 33 approximation ratio and moves each element a constant number of times in the worst case as a result of the arrival of a new input element the first algorithm moves no more than three elements while the second moves as many as seven elements please note that the number of movements performed is explicitly considered in the complexity analysis both algorithms are below the theoretical 1 536 lower bound effective for the online bin packing algorithms without the movement of elements moreover our algorithms are more online than any other linear space online bin packing algorithm because unlike the algorithms already known they allow the return of a possibly relevant fraction of bins before the work is carried out

bin packing with general cost structures following the work of anily et al we consider a variant of bin packing called bin packing with general cost structures gcbp and design an asymptotic fully polynomial time approximation scheme afptas for this problem in the classic bin packing problem a set of one dimensional items is to be assigned to subsets of total size at most 1 that is to be packed into unit sized bins however in gcbp the cost of a bin is not 1 as in classic bin packing but it is a non decreasing and concave function of the number of items packed in it where the cost of an empty bin is zero the construction of the afptas requires novel techniques for dealing with small items which are developed in this work in addition we develop a fast approximation algorithm which acts identically for all non decreasing and concave functions and has an asymptotic approximation ratio of 1 5 for all functions simultaneously

bin packing problems with rejection penalties and their dual problems in this paper we consider the following problems we are given a set of n items u1 un and a number of unit capacity bins each item ui has a size wi 0 1 and a penalty pi 0 an item can be either rejected in which case we pay its penalty or put into one bin under the constraint that the total size of the items in the bin is no greater than 1 no item can be spread into more than one bin the objective is to minimize the total number of used bins plus the total penalty paid for the rejected items we call the problem bin packing with rejection penalties and denote it as bpr for the on line bpr problem we present an algorithm with an absolute competitive ratio of 2 618 while the lower bound is 2 343 and an algorithm with an asymptotic competitive ratio arbitrarily close to 1 75 while the lower bound is 1 540 for the off line bpr problem we present an algorithm with an absolute worst case ratio of 2 while the lower bound is 1 5 and an algorithm with an asymptotic worst case ratio of 1 5 we also study a closely related bin covering version of the problem in this case pi means some amount of profit if an item is rejected we get its profit or it can be put into a bin in such a way that the total size of the items in the bin is no smaller than 1 the objective is to maximize the number of covered bins plus the total profit of all rejected items we call this problem bin covering with rejection bcr for the on line bcr problem we show that no algorithm can have absolute competitive ratio greater than 0 and present an algorithm with asymptotic competitive ratio 1 2 which is the best possible for the off line bcr problem we also present an algorithm with an absolute worst case ratio of 1 2 which matches the lower bound

nontrivial polydispersity exponents in aggregation models we consider the scaling solutions of smoluchowski s equation of irreversible aggregation for a non gelling collision kernel the scaling mass distribution f s diverges as s tau when s 0 tau is non trivial and could until now only be computed by numerical simulations we develop here new general methods to obtain exact bounds and good approximations of tau for the specific kernel kdd x y x 1 d y 1 d d describing a mean field model of particles moving in d dimensions and aggregating with conservation of mass s r d r is the particle radius perturbative and nonperturbative expansions are derived for a general kernel we find exact inequalities for tau and develop a variational approximation which is used to carry out the first systematic study of tau d d for kdd the agreement is excellent both with the expansions we derived and with existing numerical values finally we discuss a possible application to 2d decaying turbulence

optimal online algorithms for multidimensional packing problems we solve an open problem in the literature by providing an online algorithm for multidimensional bin packing that uses only bounded space to achieve this we introduce a new technique for classifying the items to be packed we show that our algorithm is optimal among bounded space algorithms for any dimension d 1 its asymptotic performance ratio is pi infty d where pi infty approx1 691 is the asymptotic performance ratio of the one dimensional algorithm harm a modified version of this algorithm for the case where all items are hypercubes is also shown to be optimal its asymptotic performance ratio is sublinear in d furthermore we extend the techniques used in these algorithms to give optimal algorithms for online bounded space variable sized packing and resource augmented packing

bayesian focalization quantifying source localization with environmental uncertainty this paper applies a bayesian formulation to study ocean acoustic source localization as a function of uncertainty in environmental properties water column and seabed and of data information content signal to noise ratio snr and number of frequencies the approach follows that of the optimum uncertain field processor a m richardson and l w nolte j acoust soc am 89 2280 2284 1991 in that localization uncertainty is quantified by joint marginal probability distributions for source range and depth integrated over uncertain environmental properties the integration is carried out here using metropolis gibbs sampling for environmental parameters and heat bath gibbs sampling for source location to provide efficient sampling over complicated parameter spaces the approach is applied to acoustic data from a shallow water site in the mediterranean sea where previous geoacoustic studies have been carried out it is found that reliable localization requires a sufficient combination of prior e

a finite version of smoluchowski s coagulation equation the authors formulate a finite version of smoluchowski s coagulation equation in which molecules of size n react in the normal way whereas molecules of size n to 2n can be produced but cannot react they prove that the solution converges to the solution of the infinite system of equations in the limit n to infinity

on bin packing with conflicts we consider the offline and online versions of a bin packing problem called bin packing with conflicts given a set of items v 1 2 dots n with sizes s 1 s 2 dots s n in 0 1 and a conflict graph g v e the goal is to find a partition of the items into independent sets of g where the total size of items in each independent set is at most 1 so that the number of independent sets in the partition is minimized this problem is clearly a generalization of both the classical one dimensional bin packing problem where e emptyset and of the graph coloring problem where s i 0 for all i 1 2 dots n since coloring problems on general graphs are hard to approximate following previous work we study the problem on specific graph classes for the offline version we design improved approximation algorithms for perfect graphs and other special classes of graphs these are a frac52 2 5 approximation algorithm for perfect graphs a frac73 approx2 33333 approximation algorithm for a subclass of perfect graphs which contains interval graphs and chordal graphs and a frac74 1 75 approximation for algorithm bipartite graphs for the online problem on interval graphs we design a 4 7 competitive algorithm and show a lower bound of frac 155 36 approx4 30556 on the competitive ratio of any algorithm to derive the last lower bound we introduce the first lower bound on the asymptotic competitive ratio of any online bin packing algorithm with known optimal value which is frac 47 36 approx1 30556

on the online bin packing problem a new framework for analyzing online bin packing algorithms is presented this framework presents a unified way of explaining the performance of algorithms based on the harmonic approach within this framework it is shown that a new algorithm harmonic has asymptotic performance ratio at most 1 58889 it is also shown that the analysis of harmonic 1 presented in richey 1991 is incorrect this is a fundamental logical flaw not an error in calculation or an omitted case the asymptotic performance ratio of harmonic 1 is at least 1 59217 thus harmonic provides the best upper bound for the online bin packing problem to date

bounds for online bin packing with cardinality constraints abstract we study a bin packing problem in which a bin can contain at most k items of total size at most 1 where k 2 is a given parameter items are presented one by one in an online fashion we analyze the best absolute competitive ratio of the problem and prove tight bounds of 2 for any k 4 additionally we present bounds for relatively small values of k with respect to the asymptotic competitive ratio and the absolute competitive ratio in particular we provide tight bounds on the absolute competitive ratio of first fit for k 2 3 4 and improve the known lower bounds on asymptotic competitive ratios for multiple values of k our method for obtaining a lower bound on the asymptotic competitive ratio using a certain type of an input is general and we also use it to obtain an alternative proof of the known lower bound on the asymptotic competitive ratio of standard online bin packing

rate distortion bounds on passive sonar performance information theory provides a novel perspective on passive sonar performance analysis this approach begins by partitioning the search space and then considers the problem of assigning a source to the correct partition based on pressure observations from a hydrophone array prior work described necessary conditions for achieving arbitrarily small probability of error pe as a tradeoff between signal to noise ratio snr and the range accuracy of the partitions this paper presents a method to extend these results using the rate distortion function to find necessary conditions for any pe the gaussian channel bound sets an upper limit on the information rate received at the array through the rate distortion function this upper limit on the information rate implies a lower bound on pe for a given partition furthermore the current work describes a tradeoff between range accuracy pe and snr examples of this tradeoff are evaluated for both one dimensional bearing and two dimensional range depth estimation problems

selfish bin packing with cardinality constraints bin packing with cardinality constraints is a variant of bin packing in this problem items with sizes of at most 1 are to be partitioned or packed into subsets called bins such that the total size of items packed into a bin would not exceed 1 and each bin would contain at most k items for a given integer parameter k 1 we consider a class of games resulting from this problem by seeing the items as selfish players trying to be packed into a valid bin where the total size of items is maximized such games always admit pure nash equilibria we analyze the price of anarchy poa of such games defined as the asymptotic worst case ratio between the maximum number of bins in a pure nash equilibrium ne and the minimum number of bins in any valid solution that is the number of bins in a socially optimal solution we provide a complete analysis of the poa as a function of k we prove that for k 2 any ne is an optimal solution for k 4 the poa is exactly 2 1k and for k 3 the poa is exactly 117

existence of gelling solutions for coagulation fragmentation equations we study the smoluchowski coagulation fragmentation equation which is an infinite set of non linear ordinary differential equations describing the evolution of a mono disperse system of particles in a well stirred solution approximating the solutions of the smoluchowski equations by a sequence of finite markov chains we investigate the qualitative behavior of the solutions we determine a device on the finite chains which can detect the gelation phenomena the density dropping phenomena it shows how the gelation phenomena are reflected on the sequence of finite markov chains using this device we determine various types of gelation kernels and get the bounds of gelation times

the discrete coagulation fragmentation equations existence uniqueness and density conservation the discrete coagulation fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones these models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys our results relate to existence uniqueness density conservation and continuous dependence and they generalise the corresponding results in ref 2 for the becker doring equations for which the processes are restricted to clusters gaining or shedding one particle examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation fragmentation equations

an efficient stochastic algorithm for studying coagulation dynamics and gelation phenomena a new class of efficient stochastic algorithms for the numerical treatment of coagulation processes is proposed the algorithms are based on the introduction of fictitious jumps combined with an acceptance rejection technique for distributions depending on particle size the increased efficiency is demonstrated by numerical experiments in particular gelation phenomena are studied

online bin packing with arbitrary release times we study a new variant of the online bin packing problem in which each item a i is associated with a size a i and also a release time r i so that it must be placed at least r i above the bottom of a bin items arrive in turn and must be assigned without any knowledge of subsequent items the goal is to pack all items into unit size bins using the minimum number of bins we study the problem with all items have equal size first we show that the any fit algorithm cannot be approximated within any constant then we present a best possible online algorithm with asymptotic competitive ratio of two

fully dynamic bin packing revisited we consider the fully dynamic bin packing problem where items arrive and depart in an online fashion and repacking of previously packed items is allowed the goal is of course to minimize both the number of bins used as well as the amount of repacking a recently introduced way of measuring the repacking costs at each timestep is the migration factor defined as the total size of repacked items divided by the size of an arriving or departing item concerning the trade off between number of bins and migration factor if we wish to achieve an asymptotic competitive ratio of 1 epsilon for the number of bins a relatively simple argument proves a lower bound of omega 1 epsilon for the migration factor we establish a nearly matching upper bound of o 1 epsilon 4 log 1 epsilon using a new dynamic rounding technique and new ideas to handle small items in a dynamic setting such that no amortization is needed the running time of our algorithm is polynomial in the number of items nand in 1 epsilon the previous best trade off was for an asymptotic competitive ratio of 5 4 for the bins rather than 1 epsilon and needed an amortized number of o log n repackings while in our scheme the number of repackings is independent of n and non amortized

a robust aptas for the classical bin packing problem bin packing is a well studied problem which has many applications in this paper we design a robust aptas for the problem the robust aptas receives a single input item to be added to the packing at each step it maintains an approximate solution throughout this process by slightly adjusting the solution for each new item at each step the total size of items which may migrate between bins must be bounded by a constant factor times the size of the new item we show that such a property cannot be maintained with respect to optimal solutions

online algorithms for 1 space bounded 2 dimensional bin packing and square packing in this paper we study 1 space bounded 2 dimensional bin packing and square packing a sequence of rectangular items square items arrive one by one each item must be packed into a square bin of unit size on its arrival without any information about future items when packing items 90 rotation is allowed 1 space bounded means there is only one active bin if the active bin cannot accommodate the coming item it will be closed and a new bin will be opened the objective is to minimize the total number of bins used for packing all items in the sequence our contributions are as follows for 1 space bounded 2 dimensional bin packing we propose an online packing algorithm with a tight competitive ratio of 5 06 a lower bound of 3 17 on the competitive ratio is proven moreover we study 1 space bounded square packing where each item is a square with side length no more than 1 a 4 3 competitive algorithm is achieved and a lower bound of 2 94 on the competitive ratio is given all these bounds surpass the previously best known results

proof of dynamical scaling in smoluchowski s coagulation equation with constant kernel smoluchowski s coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version open image in new window wherect c1 t is the concentration ofl particle clusters at timet we prove that for initial data satisfyingc1 0 0 and the condition 0 cl 0 0 the solutions behave asymptotically likec1 t t 2 c lt 1 ast withlt 1 kept fixed the scaling function c is 1 gr where rho sum l lc l 0 a conserved quantity is the initial number of particles per unit volume an analous result is obtained for the continuous version of smoluchowski s coagulation equation frac partial partial t c v text t int 0 v du text c v u text t text c u text t 2c v text t int 0 infty du text c u text t wherec v t is the oncentration of clusters of sizev

cluster size distribution in irreversible aggregation at large times it is assumed that the size distribution ck t satisfies smoluchowski s coagulation equation with rate coefficients k i j behaving as k i j approximately imu jnu i 0 the general solution ck t approaches for t to infinity the exact solution cbk t k 1 2 where the bk s are independent of the initial conditions ck 0 and can be determined from a recursion relation in class ii systems mu 0 ck t c1 t to bk t to infinity k 1 2 but the bk s depend on ck 0 only in the scaling limit k to infinity s t to infinity with k s t finite s t is the mean cluster size does ck t approach a form independent of the initial distribution class iii where ck t c1 t to infinity t to infinity k 2 3 has not been considered here

class constrained bin packing revisited we study the following variant of the bin packing problem we are given a set of items where each item has a non negative size and a color we are also given an integer parameter k and the goal is to partition the items into a minimum number of subsets such that for each subset s in the solution the total size of the items in s is at most 1 as in the classical bin packing problem and the total number of colors of the items in s is at most k which distinguishes our problem from the classical version we follow earlier work on this problem and study the problem in both offline and online scenarios

approximate strip packing in this paper we establish an algorithmic framework between bin packing and strip packing with which strip packing can be very well approximated by applying some bin packing algorithms more precisely we obtain the following results 1 any off line bin packing algorithm can be applied to strip packing maintaining almost the same asymptotic worst case ratio 2 a class of harmonic based algorithms for bin packing such as refined harmonic modified harmonic harmonic can be applied to online strip packing in particular we show that online strip packing admits an upper bound of 1 58889 on the asymptotic competitive ratio for any arbitrarily small 0 this significantly improves the previously best bound of 1 6910 and affirmatively answers an open question posed by csirik and woeginger 1997 moreover the time complexity mainly depends on a sorting procedure and the bin packing algorithms employed

new lower bounds for certain classes of bin packing algorithms on line algorithms have been extensively studied for the one dimensional bin packing problem in this paper we investigate two classes of one dimensional bin packing algorithms and we give better lower bounds for their asymptotic worst case behavior for on line algorithms so far the best lower bound was given by van vliet in 1992 12 he proved that there is no on line bin packing algorithm with better asymptotic performance ratio than 1 54014 in this paper we give an improvement on this bound to 248161 1 54037 and we investigate the parametric case as well for those lists where the elements are preprocessed according to their sizes in non increasing order csirik et al 1983 1 proved that no on line algorithm can have an asymptotic performance ratio smaller than 87 we improve this result to 5447

the class constrained bin packing problem with applications to video on demand in this paper we present approximation results for the class constrained bin packing problem that has applications to video on demand systems in this problem we are given bins of size b with c compartments and n items of q different classes each item i 1 n with class c i and size s i the problem is to pack the items into bins where each bin contains at most c different classes and has total items size at most b we present several approximation algorithms for offline and online versions of the problem

bin packing with controllable item sizes we consider a natural resource allocation problem in which we are given a set of items where each item has a list of pairs associated with it each pair is a configuration of an allowed size for this item together with a nonnegative penalty and an item can be packed using any configuration in its list the goal is to select a configuration for each item so that the number of unit bins needed to pack the sizes plus the sum of penalties is minimized this problem has applications in operating systems bandwidth allocation and discrete time cost tradeoff planning problems for the offline version of the problem we design an augmented asymptotic ptas that is an asymptotic approximation scheme that uses bins of size slightly larger than 1 we further consider the online bounded space variant where only a constant number of bins can be open simultaneously we design a sequence of algorithms where the sequence of their competitive ratios tends to the best possible asymptotic competitive ratio finally we study the bin covering problem in which a bin is covered if the sum of sizes allocated to it is at least 1 in this setting penalties are interpreted as profits and the goal is to maximize the sum of profits plus the number of covered bins we design an algorithm of best possible competitive ratio which is 2 we generalize our results for online algorithms and unit sized bins to the case of variable sized bins where there may be several distinct sizes of bins available for packing or covering and get that the competitive ratios are again the same as for the more standard online problems

a simple on line bin packing algorithm the one dimensional on line bin packing problem is considered a simple o 1 space and o n time algorithm called harmonic m is presented it is shown that this algorithm can achieve a worst case performance ratio of less than 1 692 which is better than that of the o n space and o n log n time algorithm first fit also shown is that 1 691 is a lower bound for all 0 1 space on line bin packing algorithms finally a revised version of harmonic m an o n space and o n time algorithm is presented and is shown to have a worst case performance ratio of less than 1 636 d

on line sequential bin packing we consider a sequential version of the classical bin packing problem in which items are received one by one before the size of the next item is revealed the decision maker needs to decide whether the next item is packed in the currently open bin or the bin is closed and a new bin is opened if the new item does not fit it is lost if a bin is closed the remaining free space in the bin accounts for a loss the goal of the decision maker is to minimize the loss accumulated over n periods we present an algorithm that has a cumulative loss not much larger than any strategy in a finite class of reference strategies for any sequence of items special attention is payed to reference strategies that use a fixed threshold at each step to decide whether a new bin is opened some positive and negative results are presented for this case

approximation of coalescence integrals in population balance models with local mass conservation the solution of population balance equations is a function f t r x describing a population density of particles of the property x at time t and space r for instance the additional independent variable x may denote the mass of the particle the describing equation contains additional sink and source terms involving integral operators since the coordinate x adds at least one further dimension to the spatial directions and time coordinate an efficient numerical treatment of the integral terms is crucial one of the more involved integral terms appearing in population balance models is the coalescence integral which is of the form int 0 x kappa x y y f y f x y mathrm d y the discretisation may use a locally refined grid in this paper we describe an algorithm which i is efficient the cost is mathcal o n log n n data size and ii ensures local mass conservation