stochastic analysis of a modified first fit decreasing packing we make a stochastic analysis of a modified version mffd of first fit decreasing in which each bin is closed after it receives its first fallback item consider a probability measure on 0 1 and independent random variables x1 xn distributed according to let rn rx1 xn be the number of unit size bins that mffd needs to pack items of size x1 xn we prove that c limn ern n exists and that the random variable rn nc n converges in distribution the main tools are deterministic inequalities concerning mffd that might be of independent interest

a new approach to boundary value equations and application to a generalization of airy s equation abstract the decomposition method is demonstrated to provide a convenient and computationally simple solution of equations such as 2 u ku for given boundary conditions and can easily be adapted to solution of nonlinear versions and for complex linear nonlinear or even random or coupled boundary conditions

a mathematical model of adomian polynomials the adomian decomposition method is reviewed and a mathematical model of adomian polynomials is introduced to show the convenience of this model for computer programming the mathematica package is used to generate the adomian polynomials for any order

optimal bin packing with items of random sizes consider a probability on 0 1 and i i d random variables x1 x2 xn distributed like let qn denote the optimal minimum number of unit size bins needed to pack items of size x1 x2 xn we characterize the class of which have the property that limn qn n e x1 a s or equivalently that the expected level of occupancy of bins converges to one

probabilistic analysis of a generalized bin packing problem and applications we give a unified probabilistic analysis for a general class of bin packing problems by directly analyzing corresponding mathematical programs in this general class of packing problems objects are described by a given number of attribute values some attributes may be discrete others may be continuous bins are sets of objects and the collection of feasible bins is merely required to satisfy some general consistency properties we characterize the asymptotic optimal value as the value of an easily specified linear program whose size is independent of the number of objects to be packed or as the limit of a sequence of such linear program values we also provide bounds for the rate of convergence of the average cost to its asymptotic value the analysis suggests an a s asymptotically optimal heuristic that runs in linear time the heuristics can be designed to be asymptotically optimal while still running in polynomial time we also show that in several important cases the algorithm has both po

on the solution of partial differential equations with specified boundary conditions abstract the decomposition method is applied to solution of partial differential equations in two and three dimensions with specified boundary conditions

probabilistic analysis of bin packing heuristics in this paper we present both a probabilistic and statistical analysis of several bin packing heuristics with mild conditions on the distribution of problem data we show that the ratio of the expected number of bins packed for these heuristics to the number of elements converges to some constant this constant can either be derived analytically or estimated through simulation we analyze the variance in performance of these heuristics and we further show that as the number of elements n increases the probability that the number of bins required for these heuristics exceeds any fixed percentage of their expected performance goes to zero

localization and periodicity in unstable homotopy theory in this paper we develop a hierarchy of natural localizations of spaces called the vn periodizations for n 0 which may be used to expose and study periodic phenomena in unstable homotopy theory these vn periodizations act less radically than the corresponding homological localizations 2 and respect fibrations to a very considerable extent a major part of this paper is devoted to developing the general theory of periodizations of spaces thereby providing a foundation for the study of the vn periodization and many others some of this general theory has been developed independently by dror farjoun 13 14 and we refer the reader to his work for an alternative approach with other interesting general results during the past decade remarkable progress was made by ravenel hopkins devinatz and smith 12 16 17 33 toward a global understanding of stable periodic phenomena and we hope that the present paper will help to prepare the way for a similar understanding of unstable periodic phenomena an excellent exposition of localization and periodicity in stable homotopy theory is now available in ravenel s book 35 major features of stable homotopy are understood chromatically as manifestations of more basic periodic phenomena these phenomena belong to a hierarchy starting with those detected rationally followed by those detected in classical k theory and in the successive morava k theories most fundamentally each finite cw spectrum has an intrinsic periodicity given by a vn self map which becomes a selfequivalence after suitable localization for simplicity we describe our results in the pointed homotopy category hoo of connected cw complexes for spaces w y e hoo we call y w periodic or w local when the pointed mapping space map w y is contractible or equivalently when xt w y for t 0 as shown more generally in 3 corollary 7 2 10 or 13 there is a natural initial example x pwx of a map from x to a w periodic space pwx in hoo and we call this the w periodization or w localization of x our notation is derived from the classical example where w sn l and pwx is the n th postnikov section

the a core and a cover of a group abstract this paper provides a comprehensive investigation of the cellular approximation functor cell a g in the category of groups approximating a group g by a group a we also study related notions such as a injection a generation and a constructibility of a group g and we find several interesting connections with the schur multiplier h 2 g z our constructions are direct and are given in a slow and detailed manner

postnikov pieces and bz p homotopy theory we present a constructive method to compute the cellularization with respect to b m z p for any integer m 1 of a large class of h spaces namely all those which have a finite number of non trivial b m z p homotopy groups the pointed mapping space map b m z p x is a postnikov piece we prove in particular that the b m z p cellularization of an h space having a finite number of b m z p homotopy groups is ap torsion postnikov piece along the way we characterize the bz p r cellular classifying spaces of nilpotent groups

some distributions that allow perfect packing a probability distribution on 0 1 allows perfect packing if n items of size x 1 x n independent and identically distributed according to can be packed in unit size bins in such a way that the expected wasted space is o n a large class of distributions that allow perfect packing is exhibited as a corollary the intervals a b for which the uniform distribution on a b allows perfect packing are determined

homotopical localizations of spaces for a map f of spaces dror farjoun and the author have constructed an f localization functor where a space y is called f local when map f y is an equivalence this very general construction gives all known idempotent homotopy functors of spaces the main theorem of this paper shows that f localization functors always preserve fiber sequences of connected h spaces up to small error terms for instance the localization with respect to the n th morava k theory preserves such fiber sequences up to error terms with at most three nontrivial homotopy groups this implies for example that a k 1 homology equivalence of h spaces must induce an isomorphism of v 1 periodic homotopy groups results are also obtained on the a nullification or a periodization functors which are just the f localization functors for the maps f from spaces a to points two spaces are said to have the same nullity when they give the same nullification functors and it is shown that arbitrary sets of nullity classes have both least upper bounds and greatest lower bounds the a nullifications of nilpotent postnikov spaces are completely determined

optimal placements of flexible objects i analytical results for the unbounded case the authors consider optimal placements of two dimensional flexible elastic deformable objects the objects are discs of equal size placed within a rigid boundary the paper is divided into two parts in the first part analytical results for three types of regular periodic arrangements the hexagonal square and triangular placements are presented the regular arrangements are analyzed for rectangular boundaries and radii of discs that are small compared to the area of the placement region because in this case the influence of boundary conditions can be neglected this situation is called the unbounded case they show that for the unbounded case among the three regular placements the type of hexagonal arrangements provides the largest number of placed units for the same deformation depth furthermore it can be proved that these regular placements are not too far from the truly optimal arrangements for example hexagonal placements differ at most by the factor of 1 1 from the largest possible number of generally shaped units in arbitrary arrangements these analytical results are used as guidances for testing stochastic algorithms optimizing placements of flexible objects in the second part mainly two problems are considered the underlying physical model and a simulated annealing algorithm maximizing the number of flexible discs in equilibrium placements along with the physical model an approximate formula is derived reflecting the deformation force relationship for a large range of deformations

analytic solutions for nonlinear equations solutions obtained by the decomposition method for physical problems modeled by nonlinear differential and partial differential equations without linearization or pertubation are not only continuous but analytic as well thus the solution and all derivatives are obtained

variational iteration method for solving two point boundary value problems variational iteration method is introduced to solve two point boundary value problems numerical results demonstrate that the method is promising and may overcome the difficulty arising in adomian decomposition method

variational methods for approximating solutions of nabla 2 u x f x k x u x and generalizations the iterative procedure of guy sales brami depaux and joly cabaret gsbj for approximately solving equations of the form nabla 2 u f ku is analyzed and generalized from the viewpoint of the calculus of variations the gsbj method involves a functional of the solution and either one or two auxiliary functions which are chosen empirically so as to make the iterations converge sufficiently fast even for problems for which the corresponding neumann series diverges or converges too slowly in the case in which one of the auxiliary functions is identically unity this functional is seen to be the scale invariant form of the afline functional studied by kahan rideau roussopoulos and pomraning krrp and a systematic way of seeking appropriate auxiliary functions is developed by demanding that the functional be stationary about the true solution of the equation in question the variational approach is tested on three simple example problems and the numerical results are seen to compare favorably

nullification and cellularization of classifying spaces of finite groups in this note we discuss the effect of the bz p nullification p bz p and the bz p cellularization cw bz p over classifying spaces of finite groups and we relate them with the corresponding functors with respect to moore spaces that have been intensively studied in the last years we describe p bz p bg by means of a covering fibration and we classify all finite groups g for which bg is bz p cellular we also carefully study the analogous functors in the category of groups and their relationship with the fundamental groups of p bz p bg and cw bz p bg

optimal bin packing of items of sizes uniformly distributed over 0 1 consider n independent random variables uniformly distributed over 0 1 and consider the minimum number bk of unit size bins needed to pack these items under the restriction that no bin can accept more than k items we prove that eb2 eb3 is of order n but in sharp contrast we show that eb3 ebn is of a smaller order

central extensions and generalized plus constructions we describe the effect of homological plus constructions on the homotopy groups of eilenberg maclane spaces in terms of universal central extensions

homotopy classification of self maps of bg via g actions to complete the classification of self maps of bg theorem 4 2 it remains to prove theorem 4 1 i e to show that the higher limits of certain functors n 1h 2 f p g z p wall vanish and that h0 mp g f i has the expected value in this section these limits h mp g hi will be calculated in the easy cases when g is a simply connected classical group i e one of the groups su n sp n or spin n or when i 2 the computation of h mp g hi for the simply connected exceptional lie groups will be put off until section 6 we begin with some general properties of higher limits let 6 be any small discrete category it is convenient when dealing with higher limits of functors on to let 4 moldenote the category of contravariant functors from v to ad then for any f e w muok we can identify

idempotent transformations of finite groups we describe the action of idempotent transformations on finite groups we show that finiteness is preserved by such transformations and enumerate all possible values such transformations can assign to a fixed finite simple group this is done in terms of the first two homology groups we prove for example that except special linear groups such an orbit can have at most 7 elements we also study the action of monomials of idempotent transformations on finite groups and show for example that orbits of this action are always finite

optimal bin packing with items of random sizes iii consider a probability measure mu on 0 1 and independent random variables x 1 cdots x n distributed according to mu let q n q n x 1 cdots x n be the minimum number of unit size bins needed to pack items of size x 1 cdots x n let c mu lim n to infty e q n n in this paper it is proved that the random variable q n nc mu sqrt n converges in distribution the limit is identified as a distribution of the supremum of a certain gaussian process canonically attached to mu

dual bin packing with items of random sizes given a collection of items and a number of unit size bins the dual bin packing problem requires finding the largest number of items that can be packed in these bins in our stochastic model the item sizesx1 x n are independent identically distributed according to a given probability measure denote byn n n n x1 x n the largest number of these items that can be packed in an bins where 0 a 1 is a constant we show thatb limn e n n n exists and that the random variable n n nb sqrt n converges in distribution the limit is identified as the distribution of the supremum of a certain gaussian process cannonically attached to