a blow up criterion for two dimensional compressible viscous heat conductive flows abstract we establish a blow up criterion in terms of the upper bound of the density and temperature for the strong solution to 2d compressible viscous heat conductive flows the initial vacuum is allowed

using dual approximation algorithms for scheduling problems theoretical and practical results the problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well studied problem in the theory of approximation algorithms for np hard optimization problems in this paper the strongest possible type of result for this problem a polynomial approximation scheme is presented more precisely for each e an algorithm that runs in time o n e 1 e 2 and has relative error at most e is given in addition more practical algorithms for e 1 5 2 k and e 1 6 2 k which have running times o n k log n and o n km 4 log n are presented the techniques of analysis used in proving these results are extremely simple especially in comparison with the baroque weighting techniques used previously the scheme is based on a new approach to constructing approximation algorithms which is called dual approximation algorithms where the aim is to find superoptimal but infeasible solutions and the performance is measured by the degree of infeasibility allowed this notion should find wide applicability in its own right and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive

strong convergence to global solutions for multidimensional flows of compressible viscous fluids with polytropic equations of state and discontinuous initial data we extend to general polytropic pressures p k 1 the existence theory of 8 for isothermal 1 flows of navier stokes fluids in two and three space dimensions with fairly general initial data specifically we require that the initial density be close to a constant in l2 and l and that the initial velocity be small in l2 and bounded in l2n in two dimensions the l2 norms must be weighted slightly solutions are obtained as limits of approximate solutions corresponding to mollified initial data the key point is that the approximate densities are shown to converge strongly so that nonlinear pressures can be accommodated even in the absence of any uniform regularity information for the approximate densities

existence of solutions to a model for sparse one dimensional fluids abstract we prove the global existence of solutions to a model for a viscous compressible barotropic fluid initially occupying a general open subset of a finite one dimensional interval the fluid equations are applied only on the support of the density understood in the sense of distributions this support must be tracked and accommodation must be made for the possibly infinite number of collisions of fluid packets occurring on a possibly dense set of collision times our approach avoids certain nonphysical properties of solutions which are constructed as limits of solutions in which artificial mass has been inserted

robust dimension free isoperimetry in gaussian space we prove the first robust dimension free isoperimetric result for the standard gaussian measure n and the corresponding boundary measure n in rn the main result in the theory of gaussian isoperimetry proven in the 1970s by sudakov and tsirelson and independently by borell states that if n a 1 2 then the surface area of a is bounded by the surface area of a half space with the same measure n a 2 1 2 our results imply in particular that if a rn satisfies n a 1 2 and n a 2 1 2 then there exists a half space b rn such that n a b clog 1 2 1 for an absolute constant c since the gaussian isoperimetric result was established only recently a robust version of the gaussian isoperimetric result was obtained by cianchi et al who showed that n a b c n for some function c n with no effective bounds compared to the results of cianchi et al our results have optimal i e no dependence on the dimension but worse dependence on

on spherically symmetric solutions of the compressible isentropic navier stokes equations we prove the global existence of weak solutions to the cauchy problem for the compressible isentropic navier stokes equations in n n 2 3 when the cauchy data are spherically symmetric the proof is based on the exploitation of the one dimensional feature of symmetric solutions and use of a new multidimensional property induced by the viscous flux the present paper extends lions existence theorem 15 to the case 1 n for spherically symmetric initial data where is the specific heat ratio in the pressure n 3 2 for n 2 and n 9 5 for n 3 dedicated to professor rolf leis on the occasion of his 70th birthday

global weak solutions to a two dimensional compressible mhd equations of viscous non resistive fluids abstract we consider a two dimensional mhd model describing the evolution of viscous compressible and electrically conducting fluids under the action of vertical magnetic field without resistivity existence of global weak solutions is established for any adiabatic exponent 1 inspired by the approximate scheme proposed in 15 we consider a two level approximate system with artificial diffusion and pressure term at the first level we prove global well posedness of the regularized system and establish uniform in e estimates to the regular solutions at the second level we show global existence of weak solutions to the system with artificial pressure by sending e to 0 and deriving uniform in estimates then global weak solution to the original system is constructed by vanishing the key issue in the limit passage is the strong convergence of approximate sequence of the density and magnetic field this is accomplished by following the technique developed in 15 26 and using the new technique of variable reduction developed by vasseur et al 33 in the context of compressible two fluid model so as to handle the cross terms

the large time behavior of solutions in the critical l p framework for compressible viscous and heat conductive gas flows in this paper we are concerned with non isentropic navier stokes equations governing compressible viscous and heat conductive gases we formulate an additional regularity assumption of low frequencies in rd d 3 and then establish the sharp time weighted inequality which allows us to get the time decay estimates of global strong solutions in the lp critical besov spaces precisely it is shown that if the initial data belong to some besov space b2 s1 with s1 1 d2 s0 s0 2dp d2 then the lp norm of the critical global solutions admits the time decay t s12 d2 12 1p in particular t d2p if s1 s0 which coincides with that of the heat kernel in the lp framework in comparison with the work of danchin and xu j math fluid mech 20 1641 1665 2018 the low frequency assumption is improved such that the regularity s1 belongs to the whole range 1 d2 s0

extremal properties of half spaces for log concave distributions n w x n where m s m m is the product measure in r d s y1 1 is the n n dimensional cube in r n and the infimum is then over all borel measurable n n sets a r of measure m a g p 0 p 1 h 0 we are searching for necessary and sufficent conditions under which the n 4 infimum in 1 1 is attained at the half spaces a s x g r x f c of 1 measure p for all p and h the half spaces of this form will be called standard in these notes we give such conditions in the case where m is log concave there exists the following probabilistic consequence of the above property if z z are independent random variables on some probability space 1 n v i p with common distribution m consider a family of random variables s t indexed by a set t and formed by linear combinations n s t s a t z i i is1 where the coefficients a are arbitrary functions on t such that i n s s sup a t f 1 1 i t is1 set m s sup s t then the extremal property of the standard half spaces in t

approximation algorithms for knapsack problems with cardinality constraints abstract we address a variant of the classical knapsack problem in which an upper bound is imposed on the number of items that can be selected this problem arises in the solution of real life cutting stock problems by column generation and may be used to separate cover inequalities with small support within cutting plane approaches to integer linear programs we focus our attention on approximation algorithms for the problem describing a linear storage polynomial time approximation scheme ptas and a dynamic programming based fully polynomial time approximation scheme fptas the main ideas contained in our ptas are used to derive ptas s for the knapsack problem and its multi dimensional generalization which improve on the previously proposed ptas s we finally illustrate better ptas s and fptas s for the subset sum case of the problem in which profits and weights coincide

on a parabolic hyperbolic chemotaxis system with discontinuous data well posedness stability and regularity abstract the global dynamics and regularity of parabolic hyperbolic systems is an interesting topic in pdes due to the coupling of competing dissipation and hyperbolic effects this paper is concerned with the cauchy problem of a parabolic hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis it is shown that as time tends to infinity the cauchy problem with large amplitude discontinuous data admit global weak solutions which converge to a constant state resp a viscous shock wave if the asymptotic states of initial values at far field are equal resp unequal our results improve the previous results where initial value was required to be continuous and have small amplitude numerical simulations are performed to verify our analytical results illustrate the possible regularity of solutions and speculate the minimal regularity of initial data required to obtain the smooth classical solutions of the concerned parabolic hyperbolic system

a low frequency assumption for optimal time decay estimates to the compressible navier stokes equations the global existence issue in critical spaces for compressible navier stokes equations was addressed by danchin invent math 141 579 614 2000 and then developed by charve and danchin arch rational mech anal 198 233 271 2010 chen et al commun pure appl math 63 1173 1224 2010 and haspot arch rational mech anal 202 427 460 2011 in more general lp setting the main aim of this paper is to exhibit more precisely time decay estimates of solutions constructed in the critical regularity framework to the best of our knowledge the low frequency assumption usually plays a key role in the large time asymptotics of solutions which was firstly observed by matsumura and nishida j math kyoto univ 20 67 104 1980 in the l 1 mathbb r d space we now claim a new low frequency assumption for barotropic compressible navier stokes equations which may be of interest in the mathematical analysis of viscous fluids precisely if the initial density and velocity additionally belong to some besov space dot b sigma 1 2 infty mathbb r d with the regularity sigma 1 in 1 d 2 2d p d 2 then a sharp time weighted inequality including enough time decay information can be available where optimal decay exponents for the high frequencies are exhibited the proof mainly depends on some non standard besov product estimates as a by product those optimal time decay rates of lq lr type are also captured in the critical framework

existence of global weak solutions for 3d degenerate compressible navier stokes equations in this paper we prove the existence of global weak solutions for 3d compressible navier stokes equations with degenerate viscosity the method is based on the bresch and desjardins commun math phys 238 211 223 2003 entropy conservation the main contribution of this paper is to derive the mellet and vasseur commun partial differ equ 32 431 452 2007 type inequality for weak solutions even if it is not verified by the first level of approximation this provides existence of global solutions in time for the compressible barotropic navier stokes equations the result holds for any gamma 1 in two dimensional space and for 1 gamma 3 in three dimensional space in both case with large initial data possibly vanishing on the vacuum this solves an open problem proposed by lions mathematical topics in fluid mechanics vol 2 compressible models 1998

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

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

offline black and white bin packing we define and study a variant of bin packing called unrestricted black and white bin packing similarly to standard bin packing a set of items of sizes in 0 1 are to be partitioned into subsets of total size at most 1 called bins items are of two types called black and white and the item types must alternate in each bin that is two items of the same type cannot be assigned consecutively into a bin thus a subset of items of total size at most 1 can form a valid bin if and only if the absolute value of the difference between the numbers of black items and white items in the subset is at most 1 we study this problem both with respect to the absolute and the asymptotic approximation ratios we design a fast heuristic whose absolute approximation ratio is 2 we also design an aptas and modify it into an afptas the aptas can be used as an algorithm of absolute approximation ratio 3 2 which is the best possible absolute approximation ratio for the problem unless p np

existence and uniqueness of low energy weak solutions to the compressible 3d magnetohydrodynamics equations abstract we prove the existence and uniqueness of weak solutions of the three dimensional compressible magnetohydrodynamics mhd equations we first obtain the existence of weak solutions with small l 2 norm which may display codimension one discontinuities in density pressure magnetic field and velocity gradient the weak solutions we consider here exhibit just enough regularity and structure which allow us to develop uniqueness and continuous dependence theory for the compressible mhd equations our results generalise and extend those for the intermediate weak solutions of compressible navier stokes equations

hyperbolic compressible navier stokes equations abstract we consider the non isentropic compressible navier stokes equations with hyperbolic heat conduction and a law for the stress tensor which is modified correspondingly by maxwell s law these two relaxations turning the whole system into a hyperbolic one are not only treated simultaneously but are also considered in a version having galilean invariance for this more complicated relaxed system the global well posedness is proved for small data moreover for vanishing relaxation parameters the solutions are shown to converge to solutions of the classical system

interior discontinuity for a stationary compressible stokes system with inflow datum abstract in this paper we show an interior discontinuity and its piecewise regularity for a stationary compressible stokes problem with inflow jump datum the discontinuity is preserved along the interior curve directed by the ambient vector field the difficulty lies in controlling the gradient of the pressure across the interior curve starting at the jump point that the inflow datum has we deal with this issue by constructing a vector function k corresponding to the interior jump and splitting the vector k from the velocity vector finally the remainder for the velocity vector and pressure function is shown to be in the space h 2 q h 1 q for q 2

existence of global strong solutions in critical spaces for barotropic viscous fluids this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n 2 we address the question of the global existence of strong solutions for initial data close to a constant state having critical besov regularity first this article shows the recent results of charve and danchin arch ration mech anal 198 1 233 271 2010 and chen et al commun pure appl math 63 1173 1224 2010 with a new proof our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity which allows us to cancel out the coupling between the density and the velocity as in haspot well posedness in critical spaces for barotropic viscous fluids 2009 second we improve the results of charve and danchin 2010 and chen et al 2010 by adding as in charve and danchin 2010 some regularity on the initial data in low frequencies in this case we obtain global strong solutions for a class of large initial data which rely on the results of hoff arch rational mech anal 139 303 354 1997 hoff commun pure appl math 55 11 1365 1407 2002 and hoff j math fluid mech 7 3 315 338 2005 and those of charve and danchin 2010 and chen et al 2010 we conclude by generalizing these results for general viscosity coefficients

a multi fluid compressible system as the limit of weak solutions of the isentropic compressible navier stokes equations this paper mainly concerns the mathematical justification of a viscous compressible multi fluid model linked to the baer nunziato model used by engineers see for instance ishii thermo fluid dynamic theory of two phase flow eyrolles paris 1975 under a stratification assumption more precisely we show that some approximate finite energy weak solutions of the isentropic compressible navier stokes equations converge on a short time interval to the strong solution of this viscous compressible multi fluid model provided the initial density sequence is uniformly bounded with corresponding young measures which are linear convex combinations of m dirac measures to the authors knowledge this provides in the multidimensional in space case a first positive answer to an open question see hillairet j math fluid mech 9 343 376 2007 with a stratification assumption the proof is based on the weak solutions constructed by desjardins commun partial differ equ 22 5 6 977 1008 1997 and on the existence and uniqueness of a local strong solution for the multi fluid model established by hillairet assuming initial density to be far from vacuum in a first step adapting the ideas from hoff and santos arch ration mech anal 188 509 543 2008 we prove that the sequence of weak solutions built by desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between and coupled with the uniform bound of the density property this allows us to use appropriate defect measures and their nice properties introduced and proved by hillairet aspects interactifs de la m ecanique des fluides phd thesis ens lyon 2005 in order to prove that the young measure associated to the weak limit is the convex combination of m dirac measures finally under a non degeneracy assumption of this combination stratification assumption this provides a multi fluid system using a weak strong uniqueness argument we prove that this convex combination is the one corresponding to the strong solution of the multi fluid model built by hillairet if initial data are equal we will briefly discuss this assumption to complete the paper we also present a blow up criterion for this multi fluid system following huang et al in serrin type criterion for the three dimensional viscous compressible flows arxiv 2010

global regularity of 2d density patches for inhomogeneous navier stokes this paper is about lions open problem on density patches lions in mathematical topics in fluid mechanics vol 1 volume 3 of oxford lecture series in mathematics and its applications clarendon press oxford university press new york 1996 whether or not inhomogeneous incompressible navier stokes equations preserve the initial regularity of the free boundary given by density patches using classical sobolev spaces for the velocity we first establish the propagation of c 1 gamma regularity with 0 gamma 1 in the case of positive density furthermore we go beyond this to show the persistence of a geometrical quantity such as the curvature in addition we obtain a proof for c 2 gamma regularity

existence and uniqueness of small energy weak solution to multi dimensional compressible navier stokes equations with large external potential force we study the 3 d compressible navier stokes equations with an external potential force and a general non decreasing pressure we prove the global in time existence of weak solutions with small energy initial data and with densities being non negative and essentially bounded a solution may have large oscillations and contain vacuum states no smallness assumption is made on the external force nor the initial perturbation in l for density initial velocity u0 is taken to be bounded in lq for some q 6 and no further regularity assumption is imposed on u0 finally we discuss the uniqueness of weak solutions

fujita kato theorem for the 3 d inhomogeneous navier stokes equations abstract in this paper we prove the global existence and uniqueness of the solution with discontinuous density for the 3 d inhomogeneous navier stokes equations if the initial data 0 u 0 l r 3 h s r 3 with s 1 2 satisfies 0 c 0 0 c 0 u 0 h 1 2 e for some small e 0 depending only on c 0 c 0 furthermore we introduce the dual method to show that if u 0 l p r 3 for p 6 5 2 the velocity satisfies the decay estimate k u t l 2 c 1 t k 2 p for t 1 k 0 1 with p 3 2 1 p 1 2

global strong solutions to radial symmetric compressible navier stokes equations with free boundary in this paper we consider the two dimensional barotropic compressible navier stokes equations with stress free boundary condition imposed on the free surface as the viscosity coefficients satisfies 2 1 we establish the existence of global strong solution for arbitrarily large spherical symmetric initial data even if the density vanishes across the free boundary in particular we show that the density is strictly positive and bounded from the above and below in any finite time if the initial density is strictly positive and the free boundary propagates along the particle path and expand outwards at an algebraic rate

an asymptotic fully polynomial time approximation scheme for bin covering in the bin covering problem there is a group l a1 an of items with sizes s ai 0 1 and the goal is to find a packing of the items into bins to maximize the number of bins that receive items of total size at least 1 this is a dual problem to the classical bin packing problem in this paper we present the first asymptotic fully polynomial time approximation scheme for the problem

global solutions of the navier stokes equations for multidimensional compressible flow with discontinuous initial data abstract we prove the global existence of weak solutions of the navier stokes equations for compressible isothermal flow in two and three space dimensions when the initial density is close to a constant in l 2 and l and the initial velocity is small in l 2 and bounded in l 2 n in two dimensions the l 2 norms must be weighted slightly a great deal of qualitative information about the solution is obtained for example we show that the velocity and vorticity are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a crucial role in the entire analysis particularly in closing the required energy estimates understanding rates of regularization near the initial layer and most important obtaining time independent pointwise bounds for the density

on the isoperimetric deficit in gauss space we prove a sharp quantitative version of the isoperimetric inequality in the space bbb r n endowed with the gaussian measure

a fast asymptotic approximation scheme for bin packing with rejection bin packing with rejection is the following problem given a list of items with associated sizes and rejection costs find a packing into unit bins of a subset of the list such that the number of bins used plus the sum of rejection costs of unpacked items is minimized we show that bin packing with rejection can be reduced to n multiple knapsack problems and based on techniques for the multiple knapsack problem we give a fast asymptotic polynomial time approximation scheme reject pack with time complexity o n o e 2 this improves a recent approximation scheme given by epstein which has time complexity o n o e 4 e 1 we also show that reject pack can be extended to variable sized bin packing with rejection and give an asymptotic polynomial time approximation scheme

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

quantitative isoperimetric inequalities for log convex probability measures on the line abstract the purpose of this paper is to analyze the isoperimetric inequality for symmetric log convex probability measures on the line using geometric arguments we first re prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals a result due to bobkov and houdre then we give a quantitative form of the isoperimetric inequality leading to a somehow anomalous behavior indeed it could be that a set is very close to be optimal in the sense that the isoperimetric inequality is almost an equality but at the same time is very far in the sense of the symmetric difference between sets from any extremal sets from the results on sets we derive quantitative functional inequalities of weak cheeger type

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

lagrangian structure for two dimensional non barotropic compressible fluids abstract we study the lagrangian structure for weak solutions of two dimensional navier stokes equations for a non barotropic compressible fluid i e we show the uniqueness of particle trajectories for two dimensional compressible fluids including the energy equation our result extends partially the result obtained for barotropic fluids by hoff and santos 10

global existence and asymptotic behavior for the 3d compressible navier stokes equations without heat conductivity in a bounded domain abstract in this paper we investigate the global existence and uniqueness of strong solutions to the initial boundary value problem for the 3d compressible navier stokes equations without heat conductivity in a bounded domain with slip boundary the global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in h 2 furthermore the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods

weak poincare inequalities and l2 convergence rates of markov semigroups abstract in order to describe l 2 convergence rates slower than exponential the weak poincare inequality is introduced it is shown that the convergence rate of a markov semigroup and the corresponding weak poincare inequality can be determined by each other conditions for the weak poincare inequality to hold are presented which are easy to check and which hold in many applications the weak poincare inequality is also studied by using isoperimetric inequalities for diffusion and jump processes some typical examples are given to illustrate the general results in particular our results are applied to the stochastic quantization of field theory in finite volume moreover a sharp criterion of weak poincare inequalities is presented for poisson measures on configuration spaces

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

a polynomial approximation scheme for scheduling on uniform processors using the dual approximation approach in this paper we present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors more specifically the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible we give a family of polynomial time algorithms a such that a delivers a solution that is within a relative error of e of the optimum the technique employed is the dual approximation approach where infeasible but superoptimal solutions for a related dual problem are converted to the desired feasible but possibly suboptimal solution

lagrangean structure and propagation of singularities in multidimensional compressible flow we study the propagation of singularities in solutions of the navier stokes equations of compressible barotropic fluid flow in two and three space dimensions the solutions considered are in a fairly broad regularity class for which initial densities are nonnegative and essentially bounded initial energies are small and initial velocities are in certain fractional sobolev spaces we show that if the initial density is bounded below away from zero in an open set v then each point of v determines a unique integral curve of the velocity field and that this system of integral curves defines a locally bi holder homeomorphism of v onto its image at each positive time this lagrangean structure is then applied to show that if the initial density has a limit at a point of such a set v from a given side of a continuous hypersurface in v then at each later time both the density and the divergence of the velocity have limits at the transported point from the corresponding side of the transported hypersurface which is also a continuous manifold if the limits from both sides exist then the rankine hugoniot conditions hold in a strict pointwise sense showing that the jump in the divergence of the velocity is proportional to the jump in the pressure this leads to a derivation of an explicit representation for the strength of the jump in the logarithm of the density from which it follows that discontinuities persist for all time convecting along fluid particle paths and in the case that the pressure is strictly increasing in density having strengths which decay exponentially in time

afptas results for common variants of bin packing a new method for handling the small items we consider two well known natural variants of bin packing and show that these packing problems admit asymptotic fully polynomial time approximation schemes afptass in bin packing problems a set of one dimensional items of size at most 1 is to be assigned packed to subsets of sum at most 1 bins it has been known for a while that the most basic problem admits an afptas in this paper we develop new methods that allow us to extend this result to other variants of bin packing consisting of a family of two parameter bin packing problems we demonstrate the use of our methods by designing afptass for the following problems the first problem is bin packing with cardinality constraints where a parameter k is given such that a bin may contain up to k items the goal is to minimize the number of bins used the second problem is bin packing with rejection where every item has a rejection penalty associated with it an item needs to be either packed to a bin or rejected and the goal is to minimize the number of bins used and the total rejection penalty of unpacked items this resolves the complexity of two important variants of the bin packing problem our approximation schemes use a novel method for packing the small items this new method is the core of the improved running times of our schemes over the running times of the previous results which are only asymptotic polynomial time approximationschemes

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

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

global existence of weak solutions to two dimensional compressible viscoelastic flows abstract the global existence of weak solutions of the compressible viscoelastic flows in two spatial dimensions is studied in this paper we show the global existence if the initial velocity u 0 is small in h with an arbitrary 0 1 2 and the perturbation of 0 f 0 around the constant state 1 i are small in l 2 b p 1 2 p with p 1 1 16 2 4 one of the main ingredients is that the velocity and the effective viscous flux g i are sufficiently regular for positive time the regularity of g i helps to obtain the l estimate of density and deformation gradient and hence eliminates the possible concentration and oscillation issues