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

using fast matrix multiplication to find basic solutions abstract we consider the problem of finding a basic solution to a system of linear constraints in standard form given a non basic solution to the system we show that the known arithmetic complexity bounds for this problem admit considerable improvement our technique which is similar in spirit to that used by vaidya to find the best complexity bounds for linear programming is based on reducing much of the computation involved to matrix multiplication consequently our complexity bounds in their most general form are a function of the complexity of matrix multiplication using the best known algorithm for matrix multiplication we achieve a running time of o m1 594 n arithmetic operations for an m n problem in standard form previously the best bound was o m2 n arithmetic operations

the spatial isomorphism problem for close separable nuclear c algebras the kadison kastler problem asks whether close c algebras on a hilbert space must be spatially isomorphic we establish this when one of the algebras is separable and nuclear we also apply our methods to the study of near inclusions of c algebras

decomposition rank and z stability we show that separable simple nonelementary unital c algebras with finite decomposition rank absorb the jiang su algebra mathcal z tensorially this has a number of consequences for elliott s program to classify nuclear c algebras by their k theory data in particular it completes the classification of c algebras associated to uniquely ergodic smooth minimal dynamical systems by their ordered k groups

fully dynamic algorithms for bin packing being mostly myopic helps the problem of maintaining an approximate solution for one dimensional bin packing when items may arrive and depart dynamically is studied in accordance with various work on fully dynamic algorithms and in contrast to prior work on bin packing it is assumed that the packing may be arbitrarily rearranged to accommodate arriving and departing items in this context our main result is a fully dynamic approximation algorithm for bin packing mmp that is frac 5 4 competitive and requires theta log n time per operation i e for an insert or a delete of an item this competitive ratio of frac 5 4 is nearly as good as that of the best practical off line algorithms our algorithm utilizes the technique introduced here whereby the packing of an item is done with a total disregard for already packed items of a smaller size this myopic packing of an item may then cause several smaller items to be repacked in a similar fashion with a bit of additional sophistication to avoid certain bad cases the number of items either individual items or bundles of very small items treated as a whole that needs to be repacked is bounded by a constant

mixed methods for the helmholtz transmission eigenvalues the ciarlet raviart mixed finite element method is popular for the biharmonic equations in this paper we apply this method to the helmholtz transmission eigenvalue problem which is quadratic and

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

dynamic bin packing motivated by potential applications to computer storage allocation we generalize the classical one dimensional bin packing model to include dynamic arrivals and departures of items over time within this setting we prove close upper and lower bounds on the worst case performance of the commonly used first fit packing algorithm and using adversary type arguments we show that no on line packing algorithm can satisfy a substantially better performance bound than that for first fit

perturbations of nuclear c algebras kadison and kastler introduced a natural metric on the collection of all c subalgebras of the bounded operators on a separable hilbert space they conjectured that sufficiently close algebras are unitarily conjugate we establish this conjecture when one algebra is separable and nuclear we also consider one sided versions of these notions and we obtain embeddings from certain near inclusions involving separable nuclear c algebras at the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme

efficient spectral galerkin methods iii polar and cylindrical geometries we present in this paper several extremely efficient and accurate spectral galerkin methods for second and fourth order equations in polar and cylindrical geometries these methods are based on appropriate variational formulations which incorporate naturally the pole condition s in particular the computational complexities of the chebyshev galerkin method in a disk and the chebyshev legendre galerkin method in a disk or a cylinder are quasi optimal optimal up to a logarithmic term as an indication of efficiency the cpu time for the poisson solver on a disk by our chebyshev galerkin method is only about 70 of the corresponding finite difference code in fishpack

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

decomposable approximations of nuclear c algebras abstract we show that nuclear c algebras have a refined version of the completely positive approximation property in which the maps that approximately factorize through finite dimensional algebras are convex combinations of order zero maps we use this to show that a separable nuclear c algebra a which is closely contained in a c algebra b embeds into b

regularity properties in the classification program for separable amenable c algebras we report on recent progress in the program to classify separable amenable c algebras our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank strict comparison of positive elements and z stability and on the importance of the cuntz semigroup we include a brief history of the program s successes since 1989 a more detailed look at the villadsen type algebras which have so dramatically changed the landscape and a collection of announcements on the structure and properties of the cuntz semigroup

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

iterative methods for transmission eigenvalues transmission eigenvalues have important applications in inverse scattering theory they can be used to obtain useful information of the physical properties such as the index of refraction of the scattering target despite considerable effort devoted to the existence and estimation for the transmission eigenvalues the numerical treatment is limited since the problem is nonstandard classical finite element methods result in non hermitian matrix eigenvalue problems in this paper we focus on the computation of a few lowest transmission eigenvalues which are of practical importance instead of a non hermitian problem we work on a series of generalized hermitian problems we first use a fourth order reformulation of the transmission eigenproblem to construct functions involving an associated generalized eigenvalue problem the roots of these functions are the transmission eigenvalues then we apply iterative methods to compute the transmission eigenvalues we show the convergence of the numerical schemes the effectiveness of the methods is demonstrated using various numerical examples

spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem we first develop an efficient spectral galerkin method and a rigorous error analysis for the generalized eigenvalue problems associated to a transmission eigenvalue problem then we present an iterative scheme based on computation of the first transmission eigenvalue to estimate the index of refraction of an inhomogeneous medium we present ample numerical results to demonstrate the effectiveness and accuracy of our approach

nuclear dimension and mathcal z stability of pure c algebras in this article i study a number of topological and algebraic dimension type properties of simple c algebras and their interplay in particular a simple c algebra is defined to be tracially m bar m pure if it has strong tracial m comparison and is tracially bar m almost divisible these notions are related to each other and to nuclear dimension

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

robust approximation schemes for cube packing square bin packing where square items are to be packed into a minimum number of unit size squares and its multidimensional generalization cube packing where d dimensional cubic items are to be packed into a minimum number of unit size cubes of the same dimension are well studied generalizations of the classical one dimensional bin packing problem these problems are known to have asymptotic polynomial time approximation schemes aptas in this paper we design robust approximation schemes for these problems at each step a robust algorithm receives a single input item to be added to the packing the new item must be packed and the packing can be modified but the total volume of items which may migrate between bins or change their positions inside bins must be bounded by a constant factor times the volume of the new item that is the solution is created by slightly adjusting the solution upon the arrival of each new item previous approximation schemes partition items into three types according

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

an efficient spectral galerkin approximation based on dimension reduction scheme for transmission eigenvalues in polar geometries abstract in this paper we put forward an efficient spectral galerkin approximation in view of dimension reduction scheme for transmission eigenvalue problem in polar geometries firstly we turn the original problem into an equivalent fourth order nonlinear eigenvalue problem then the fourth order nonlinear eigenvalue problem is transformed into a coupled fourth order linear eigenvalue system by introducing an auxiliary poisson equation secondly based on polar coordinate transformation we further reduce the coupled fourth order linear eigenvalue system to a series of equivalent one dimensional eigenvalue systems thirdly we derive the essential polar condition and introduce the appropriate weighted sobolev space according to the polar condition and establish the weak form and the corresponding discrete form in addition by utilizing spectral theory of compact operators we prove the error estimates of approximation eigenvalues and eigenvectors for each one dimensional eigenvalue system finally we provide ample numerical experiments and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results