efficient algorithms for separated continuous linear programs the multicommodity flow problem with holding costs and extensions we give an approximation scheme for separated continuous linear programming problems such problems arise as fluid relaxations of multiclass queueing networks and are used to find approximate solutions to complex job shop scheduling problems in a network with linear flow costs and linear per unit time holding costs our algorithm finds a drainage of the network that for given constants e 0 and i 0 has total cost 1 e opt i where opt is the cost of the minimum cost drainage the complexity of our algorithm is polynomial in the size of the input network 1 e and log 1 i the fluid relaxation is a continuous problem while the problem is known to have a piecewise constant solution it is not known to have a polynomially sized solution we introduce a natural discretization of polynomial size and prove that this discretization produces a solution with low cost this is the first polynomial time algorithm with a provable approximation guarantee for fluid relaxations

the invertibility of rotation invariant radon transforms let r denote the radon transform on rn that integrates a function over hyperplanes in given smooth positive measures depending on the hyperplane we characterize the measures for which r is rotation invariant we prove rotation invariant transforms are all one to one and hence invertible on the domain of square integrable functions of compact support l02 rn we prove the hole theorem if f l02rn and r f 0 for hyperplanes not intersecting a ball centered at the origin then f is zero outside of that ball using the theory of fourier integral operators we extend these results to the domain of distributions of compact support on rn our results prove invertibility for a mathematical model of positron emission tomography and imply a hole theorem for the constantly attenuated radon transform as well as invertibility for other radon transforms

a simplex based algorithm to solve separated continuous linear programs we consider the separated continuous linear programming problem with linear data we characterize the form of its optimal solution and present an algorithm which solves it in a finite number of steps using an analog of the simplex method in the space of bounded measurable functions

approximation techniques for average completion time scheduling we consider the problem of nonpreemptive scheduling to minimize average weighted completion time allowing for release dates parallel machines and precedence constraints recent work has led to constant factor approximations for this problem based on solving a preemptive or linear programming relaxation and then using the solution to get an ordering on the jobs we introduce several new techniques which generalize this basic paradigm we use these ideas to obtain improved approximation algorithms for one machine scheduling to minimize average completion time with release dates in the process we obtain an optimal randomized on line algorithm for the same problem that beats a lower bound for deterministic on line algorithms we consider extensions to the case of parallel machine scheduling and for this we introduce two new ideas first we show that a preemptive one machine relaxation is a powerful tool for designing parallel machine scheduling algorithms that simultaneously produce good approximations and have small running times second we show that a nongreedy rounding of the relaxation yields better approximations than a greedy one we also prove a general theorem relating the value of one machine relaxations to that of the schedules obtained for the original m machine problems this theorem applies even when there are precedence constraints on the jobs we apply this result to obtain improved approximation ratios for precedence graphs such as in trees out trees and series parallel graphs

mckay correspondence for symplectic quotient singularities let v be a finite dimensional complex vector space and g sl v a finite subgroup the quotient variety x v g is usually singular if dim v 2 then there exists a canonical smooth projective resolution y x of the singular variety x this resolution is crepant which in this case means that the manifold y has trivial canonical bundle we recall the precise definitions in definition 1 2 the fiber 1 0 y over the singular point 0 x v g is a rational curve whose components are numbered by the nontrivial conjugacy classes in g the homology classes of these components freely generate h2 y q all the other homology groups of y are trivial except for h0 y q q this situation was described by j mckay in mck it is known as the mckay correspondence in higher dimensions the picture splits into two parts one can consider two separate questions 1 when does the quotient x v g admit a smooth crepant resolution y x 2 assuming there exists a smooth crepant resolution y x what can one say about the homology h y q

radon cosine and sine transforms on real hyperbolic space new pointwise inversion formulae are obtained for the d dimensional totally geodesic radon transform on the n dimensional real hyperbolic space 1 d n 1 in terms of polynomials of the laplace beltrami operator and intertwining fractional integrals similar results are established for hyperbolic cosine and sine transforms

asymptotic formulas for the dual radon transform and applications caracterisation de la transformation de radon duale d une fonction indefiniment differentiable formules asymptotiques et applications

on the relationship between combinatorial and lp based lower bounds for np hard scheduling problems enumerative approaches to solving optimization problems such as branch and bound require a subroutine that produces a lower bound on the value of the optimal solution in the domain of scheduling problems the requisite lower bound has typically been derived from either the solution to a linear programming lp relaxation of the problem or the solution to a combinatorial relaxation in this paper we investigate from a theoretical perspective the relationship between several lp based lower bounds and combinatorial lower bounds for three scheduling problems in which the goal is to minimize the average weighted completion time of the jobs scheduled we establish a number of facts about the relationship between these different sorts of lower bounds including the equivalence of certain lp based lower bounds for these problems to combinatorial lower bounds used in successful branch and bound algorithms as a result we obtain the first worst case analysis of the quality of the lower bounds delivered by these combinatorial relaxations

inequalities between projection functions of convex bodies we obtain injectivity results for certain integral transforms on grassmann manifolds these are used to solve the intermediate cases of a problem of shephard and to give other applications to projection functions of convex bodies of revolution

generalized intersection bodies abstract we study the structures of two types of generalizations of intersection bodies and the problem of whether they are in fact equivalent intersection bodies were introduced by lutwak and played a key role in the solution of the busemann petty problem a natural geometric generalization of this problem considered by zhang led him to introduce one type of generalized intersection bodies a second type was introduced by koldobsky who studied a different analytic generalization of this problem koldobsky also studied the connection between these two types of bodies and noted that an equivalence between these two notions would completely settle the unresolved cases in the generalized busemann petty problem we show that these classes share many identical structural properties proving the same results using integral geometry techniques for zhang s class and fourier transform techniques for koldobsky s class using a functional analytic approach we give several surprising equivalent formulations for the equivalence problem which reveal a deep connection to several fundamental problems in the integral geometry of the grassmann manifold

on the equivalence of the max min transportation lower bound and the time indexed lower bound for single machine scheduling problems new observations are made about two lower bound schemes for single machine min sum scheduling problems we find that the strongest bound of those provided by transportation problem relaxations can be computed by solving a linear program we show the equivalence of this strongest bound and the bound provided by the lp relaxation of the time indexed integer programming formulation these observations lead to a new lower bound scheme that yields fast approximation of the time indexed bound several techniques are developed to facilitate the effective use of the new lower bound in branch and bound numerical experiments are conducted on 375 benchmark problems of the total weighted tardiness problem from or library results obtained with our new method are spectacular we are able to solve all 125 open problems to optimality

scheduling to minimize average completion time off line and on line approximation algorithms in this paper we introduce two general techniques for the design and analysis of approximation algorithms for np hard scheduling problems in which the objective is to minimize the weighted sum of the job completion times for a variety of scheduling models these techniques yield the first algorithms that are guaranteed to find schedules that have objective function value within a constant factor of the optimum in the first approach we use an optimal solution to a linear programming relaxation in order to guide a simple list scheduling rule consequently we also obtain results about the strength of the relaxation our second approach yields on line algorithms for these problems in this setting we are scheduling jobs that continually arrive to be processed and for each time t we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards our on line technique yields constant performance guarantees for a variety of scheduling environments and in some cases essentially matches the performance of our off line lp based algorithms

a new algorithm for state constrained separated continuous linear programs during the last few decades significant progress has been made in solving large scale finite dimensional and semi infinite linear programming problems in contrast little progress has been made in solving linear programs in infinite dimensional spaces despite their importance as models in manufacturing and communication systems inspired by the research on separated continuous linear programs we propose a new class of continuous linear programming problems that has a variety of important applications in communications manufacturing and urban traffic control this class of continuous linear programs contains the separated continuous linear programs as a subclass using ideas from quadratic programming we propose an efficient algorithm for solving large scale problems in this new class under mild assumptions on the form of the problem data we prove algorithmically the absence of a duality gap for this class of problems without any boundedness assumptions on the solution set we show this class of problems admits piecewise constant optimal control when the optimal solution exists we give conditions for the existence of an optimal solution we also report computational results which illustrate that the new algorithm is effective in solving large scale realistic problems with several hundred continuous variables arising in manufacturing systems

multiproduct systems with both setup times and costs fluid bounds and schedules this paper considers a multiproduct single server production system where both setup times and costs are incurred whenever the server changes product the system is make to order with a per unit backlogging cost the objective is to minimize the long run average cost per unit time using a fluid model we provide a closed form lower bound on system performance this bound is also shown to provide a lower bound for stochastic systems when scheduling is local or static but is only an approximation when scheduling is global or dynamic the fluid bound suggests both local and global scheduling heuristics which are tested for the stochastic system via a simulation study

hyperbolic geometry and local dirichlet neumann map abstract we show in three dimensions that if we know the dirichlet to neumann dn map associated to the schrodinger equation with a potential around the intersection of a sphere centered outside the convex hull of a bounded domain with smooth boundary we can determine uniquely the corresponding spherical mean of the potential with an appropriate measure this allows one to determine the potential near the boundary where the dn map is measured similar results are valid for electrical impedance tomography

worst case bound of an lrf schedule for the mean weighted flow time problem this paper studies the problem of scheduling a set of n independent tasks on m identical processors so as to minimize mean weighted flow time the problem is known to be np complete for m geqq 2 and to be np complete in the strong sense for m arbitrary the worst case behavior of a heuristic algorithm which requires time o n log n is investigated and it is shown that the mean weighted flow time obtained by the algorithm does not exceed sqrt 2 1 2 cong 1 0207 times that of an optimal schedule moreover the bound sqrt 2 1 2 is best possible

the second mixed projection problem and the projection centroid conjectures abstract we provide partial answers to the open problems 4 5 4 6 of 3 and 12 9 of 12 regarding the classification of fixed points of the second mixed projection operator and iterates of the projection and centroid operators

poisson deformations of symplectic quotient singularities abstract we establish a connection between smooth symplectic resolutions and symplectic deformations of a possibly singular affine poisson variety in particular let v be a finite dimensional complex symplectic vector space and g sp v a finite subgroup our main result says that the so called calogero moser deformation of the orbifold v g is in an appropriate sense a versal poisson deformation that enables us to determine the algebra structure on the cohomology h x c of any smooth symplectic resolution x v g multiplicative mckay correspondence we prove further that if g gl h is an irreducible weyl group and v h h then no smooth symplectic resolution of v g exists unless g is of types a b c

the radon transform on the heisenberg group and the transversal radon transform abstract the radon transform on the heisenberg group was introduced by r strichartz we regard it as a particular case of a more general transversal radon transform that integrates functions on r m over hyperplanes meeting the last coordinate axis the paper contains new boundedness results and explicit inversion formulas for both transforms of l p functions in the full range of the parameter p we also show that these transforms are isomorphisms of the corresponding semyanistyi lizorkin spaces of smooth functions in the framework of these spaces we obtain inversion formulas which are pointwise analogues of the corresponding formulas by r strichartz

improved approximation algorithms for shop scheduling problems in the job shop scheduling problem there are m machines and n jobs a job consists of a sequence of operations each of which must be processed on a specified machine and the aim is to complete all jobs as quickly as possible this problem is strongly cal np hard even for very restrictive special cases the authors give the first randomized and deterministic polynomial time algorithms that yield polylogarithmic approximations to the optimal length schedule these algorithms also extend to the more general case where a job is given not by a linear ordering of the machines on which it must be processed but by an arbitrary partial order comparable bounds can also be obtained when there are m types of machines a specified number of machines of each type and each operation must be processed on one of the machines of a specified type as well as for the problem of scheduling unrelated parallel machines subject to chain precedence constraints

harmonic analysis on grassmannian bundles the harmonic analysis of the grassmannian bundle of k dimensional affine subspaces of rn as a homogeneous space of the euclidean motion group is given explicitly this is used to obtain the diagonalization of various generalizations of the radon transform between such bundles in abstract form the same technique gives the plancherel formula for any unitary representation of a semidirect product g x v v a normal abelian subgroup induced from an irreducible unitary representation of a subgroup of the form h x w introduction let gn k denote the grassmannian manifold of linear k dimensional subspaces of rn and let pn k denote the grassmannian bundle of affine kdimensional subspaces also called k planes so pn k is a bundle over gn k with fibre dimension m where m k n in group theoretic terms gn k g h where g so n and h s o k x 0 m or we could just as well take g 0 n and h o k x 0 m let p denote the standard representation of g on v rn and form the semidirect product g x v the euclidean motion group with group law gi xi o g x gig xi p gl x the eucidean motion group acts naturally on the bundle pn k if we let wr0 denote the subspace spanned by the first k basis elements el ek then the isotropy subgroup of wro is h x w where w is the subspace 7r0 thus pn k g v h x w in this paper we obtain the explicit harmonic analysis on pn k the decomposition of l2 pn k with respect to the invariant measure into a direct integral of irreducible representations of g x v this is accomplished in two stages in the first abstract stage 2 we let g be any compact lie group and p any real representation of g on v and for functions f g x invariant under the right action of a subgroup h x w we obtain the fourier decomposition formula f g x j e fa 6 g x dv a a6g here a is an index set for the orbits in w under the action of h if we choose a point za in each orbit and let gc and ha denote the isotropy subgroup of za in g and h respectively then a x denotes the set of irreducible representations of gca that occur in l2 gca ha the measure dv a is determined by dz da dv a where dz is lebesgue measure on w and da is the invariant measure on the orbit received by the editors march 7 1985 1980 mathematics subject classification primary 43a85 secondary 22d30 44a15

scheduling parallel machines with inclusive processing set restrictions and job release times we consider the problem of scheduling a set of jobs with different release times on parallel machines so as to minimize the makespan of the schedule the machines have the same processing speed but each job is compatible with only a subset of those machines the machines can be linearly ordered such that a higher indexed machine can process all those jobs that a lower indexed machine can process we present an efficient algorithm for this problem with a worst case performance ratio of 2 we also develop a polynomial time approximation scheme ptas for the problem as well as a fully polynomial time approximation scheme fptas for the case in which the number of machines is fixed

convex quadratic and semidefinite programming relaxations in scheduling we consider the problem of scheduling unrelated parallel machines subject to release dates so as to minimize the total weighted completion time of jobs the main contribution of this paper is a provably good convex quadratic programming relaxation of strongly polynomial size for this problem the best previously known approximation algorithms are based on lp relaxations in time or interval indexed variables those lp relaxations however suffer from a huge number of variables as a result of the convex quadratic programming approach we can give a very simple and easy to analyze 2 approximation algorithm which can be further improved to performance guarantee 3 2 in the absence of release dates we also consider preemptive scheduling problems and derive approximation algorithms and results on the power of preemption which improve upon the best previously known results for these settings finally for the special case of two machines we introduce a more sophisticated semidefinite programming relaxation and apply the random hyperplane technique introduced by goemans and williamson for the maxcut problem this leads to an improved 1 2752 approximation

an algorithm for a class of continuous linear programs this paper discusses a class of continuous linear programs posed in a function space called separated continuous linear programs sclp a dual linear program and a corresponding discrete approximation are introduced followed by a discussion of their properties the discrete approximation gives rise to an improvement step which is constructed from any given feasible nonoptimal solution for sclp a strong duality result follows from this there are a variety of possible implementations of an algorithm for solving sclp problems using this improvement step finally some computational results are given from one possible implementation

asymptotic performance ratio of an online algorithm for the single machine scheduling with release dates we analyze the single machine weighted completion time problem with release dates and prove that the asymptotic performance ratio of a simple online algorithm is one this implies that this algorithm generates a solution whose relative error decreases to zero as the number of jobs increases our proof does not require any probabilistic assumption on the job parameters and relies heavily on properties of optimal solutions to a linear programming relaxation of the problem

generalized online routing new competitive ratios resource augmentation and asymptotic analyses we consider online routing optimization problems where the objective is to minimize the time needed to visit a set of locations under various constraints the problems are online because the set of locations are revealed incrementally over time we consider two main problems 1 the online traveling salesman problem tsp with precedence and capacity constraints and 2 the online tsp with m salesmen for both problems we propose online algorithms each with a competitive ratio of 2 for the m salesmen problem we show that our result is best possible we also consider polynomial time online algorithms we then consider resource augmentation where we give the online servers additional resources to offset the powerful offline adversary advantage in this way we address a main criticism of competitive analysis we consider the cases where the online algorithm has access to faster servers servers with larger capacities additional servers and or advanced information we derive improved competitive ratios we also give lower bounds on the competitive ratios under resource augmentation which in many cases are tight and lead to best possible results finally we study online algorithms from an asymptotic point of view we show that under general stochastic structures for the problem data unknown and unused by the online player the online algorithms are almost surely asymptotically optimal furthermore we provide computational results that show that the convergence can be very fast

the inversion problem and applications of the generalized radon transform we prove that under certain conditions the inversion problem for the generalized radon transform reduces to solving a fredholm integral equation and we obtain the asymptotic expansion of the symbol of the integral operator in this equation we consider applications of the generalized radon transform to partial differential equations with variable coefficients and provide a solution to the inversion problem for the attenuated and exponential radon transforms

all schatten spaces endowed with the schur product are q algebras we prove that the banach algebra formed by the space of compact operators on a hilbert space endowed with the schur product is a quotient of a uniform algebra also known as a q algebra together with a similar result of perez garcia for the trace class this completes the answer to a long standing question of varopoulos

the policy iteration algorithm for average reward markov decision processes with general state space the average cost optimal control problem is addressed for markov decision processes with unbounded cost it is found that the policy iteration algorithm generates a sequence of policies which are c regular where c is the cost function under consideration this result only requires the existence of an initial c regular policy and an irreducibility condition on the state space furthermore under these conditions the sequence of relative value functions generated by the algorithm is bounded from below and nearly decreasing from which it follows that the algorithm is always convergent under further conditions it is shown that the algorithm does compute a solution to the optimality equations and hence an optimal average cost policy these results provide elementary criteria for the existence of optimal policies for markov decision processes with unbounded cost and recover known results for the standard linear quadratic gaussian problem in particular in the control of multiclass queueing networks it is found that there is a close connection between optimization of the network and optimal control of a far simpler fluid network model

a ptas for minimizing the total weighted completion time on identical parallel machines we consider the problem of scheduling a set of n jobs on m identical parallel machines so as to minimize the weighted sum of job completion times this problem is np hard in the strong sense the best approximation result known so far was a documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document 1 over 2 1 sqrt 2 end document approximation algorithm that has been derived by kawaguchi and kyan back in 1986 the contribution of this paper is a polynomial time approximation scheme for this setting which settles a problem that was open for a long time moreover our result constitutes the first known approximation scheme for a strongly np hard scheduling problem with minsum objective

small contractions of symplectic 4 folds we classify small contractions of holomorphically symplectic 4 folds ams msc 14e30 14j35

harmonic analysis on constant curvature surfaces with point singularities let rn2 denote the n fold covering space of the plane with the origin deleted 0 n parametrized by polar coordinates r with periodic of period 2 n and equipped with the flat riemannian metric dr2 r2 d 2 there is a natural harmonic analysis with the fourier transform given by f c 12 0 f r j v r eiv r dr d fourier inversion formula f c 0 f r j v r eiv d dv and plancherel formula f 22 2 0 f v 2 d dv in the case n and analogous expressions with v restricted to the values kn when n is finite this harmonic analysis realizes the joint spectral decompositions of the self adjoint operators i and where is the dirichlet or friedrichs extension of the laplace beltrami operator defined on d this paper gives a detailed development of this harmonic analysis including the effect of basic differentiation and multiplication operators on the fourier transform the theory of l2 sobolev spaces hk and the distinctions between h2 the domain of and the space of functions f with both f and f in l2 in the process it is necessary to study certain generalizations of the hilbert transform that are related to the theory of hankel transforms two applications of this harmonic analysis are given to the study of a natural radon transform or x ray transform on rn2 and to the solution of the heat equation on rn2 the latter also gives information about the heat equation in a sector of the plane with dirichlet or neumann boundary conditions and about brownian motion in the plane taking into account the winding number of a brownian path with respect to the origin analogous problems for spaces of non zero constant curvature are discussed and in this context some higher dimensional spaces arise naturally

birational geometry of symplectic quotient singularities for a finite subgroup gamma subset mathrm sl 2 mathbb c and for n ge 1 we use variation of git quotient for nakajima quiver varieties to study the birational geometry of the hilbert scheme of n points on the minimal resolution s of the kleinian singularity mathbb c 2 gamma it is well known that x mathrm mathrm hilb n s is a projective crepant resolution of the symplectic singularity mathbb c 2n gamma n where gamma n gamma wr mathfrak s n is the wreath product we prove that every projective crepant resolution of mathbb c 2n gamma n can be realised as the fine moduli space of theta stable pi modules for a fixed dimension vector where pi is the framed preprojective algebra of gamma and theta is a choice of generic stability condition our approach uses the linearisation map from git to relate wall crossing in the space of theta stability conditions to birational transformations of x over mathbb c 2n gamma n as a corollary we describe completely the ample and movable cones of x over mathbb c 2n gamma n and show that the mori chamber decomposition of the movable cone is determined by an extended catalan hyperplane arrangement of the ade root system associated to gamma by the mckay correspondence in the appendix we show that morphisms of quiver varieties induced by variation of git quotient are semismall generalising a result of nakajima in the case where the quiver variety is smooth

on the asymptotic optimality of a simple on line algorithm for the stochastic single machine weighted completion time problem and its extensions we consider the stochastic single machine problem when the objective is to minimize the expected total weighted completion time of a set of jobs that are released over time we assume that the existence and the parameters of each job including its release date weight and expected processing times are not known until its release date the actual processing times are not known until processing is completed we analyze the performance of the on line nonpreemptive weighted shortest expected processing time among available jobs wsepta heuristic when a scheduling decision needs to be made this heuristic assigns among the jobs that have arrived but not yet processed one with the largest ratio of its weight to its expected processing time we prove that when the job weights and processing times are bounded and job processing times are mutually independent random variables wsepta is asymptotically optimal for the single machine problem this implies that wsepta generates a solution whose relative error approaches zero as the number of jobs increases this result can be extended to the stochastic flow shop and open shop problems as well as models with stochastic job weights

from fluid relaxations to practical algorithms for high multiplicity job shop scheduling the holding cost objective we design an algorithm for the high multiplicity job shop scheduling problem with the objective of minimizing the total holding cost by appropriately rounding an optimal solution to a fluid relaxation in which we replace discrete jobs with the flow of a continuous fluid the algorithm solves the fluid relaxation optimally and then aims to keep the schedule in the discrete network close to the schedule given by the fluid relaxation if the number of jobs from each type grow linearly withn then the algorithm is within an additive factoro n from the optimal which scales aso n2 thus it is asymptotically optimal we report computational results on benchmark instances chosen from the or library comparing the performance of the proposed algorithm and several commonly used heuristic methods these results suggest that for problems of moderate to high multiplicity the proposed algorithm outperforms these methods and for very high multiplicity the overperformance is dramatic for problems of low to moderate multiplicity however the relative errors of the heuristic methods are comparable to those of the proposed algorithm and the best of these methods performs better overall than the proposed method

radon transforms on grassmann manifolds if n p q 1 the affine grassmann manifolds g p n of p planes and g q n of q planes in rn are dual homogeneous spaces of the euclidean motion group m n with g p n and g q n being incident if and only if meets at a right angle the resulting integral transform from functions on g p n to functions on g q n generalizes both the classical radon transform and its dual we show that this transform intertwines the actions of certain m n invariant differential operators on g p n and g q n and we prove an inversion formula for this transform when n is odd generalizing the radon inversion formula and obtaining in particular an inversion formula for the dual transform

nonlocal inversion formulas for the x ray transform etude de la reconstruction d une fonction f x sur r n a partir de ses integrales sur des plans k dimensionnels cas de k 1

minimizing average completion time in the presence of release dates a natural and basic problem in scheduling theory is to provide good average quality of service to a stream of jobs that arrive over time in this paper we consider the problem of schedulingn jobs that are released over time in order to minimize the average completion time of the set of jobs in contrast to the problem of minimizing average completion time when all jobs are available at time 0 all the problems that we consider are np hard and essentially nothing was known about constructing good approximations in polynomial time we give the first constant factor approximation algorithms for several variants of the single and parallel machine models many of the algorithms are based on interesting algorithmic and structural relationships between preemptive and nonpreemptive schedules and linear programming relaxations of both many of the algorithms generalize to the minimization of averageweighted completion time as well 1998 the mathematical programming society inc published by elsevier science b v

a backprojection algorithm for electrical impedance imaging the authors study a two dimensional reconstruction algorithm due to d c barber and b h brown applied to a linearized electrostatic inverse problem first the authors demonstrate how this algorithm fits within the framework of inverses of generalized radon transforms studied by g beylkin second an iterative improvement of the barber brown algorithm is constructed based on the conjugate residual method several numerical results obtained with this iterative algorithm are presented

a posteriori error estimate and convergence analysis for conductivity image reconstruction in mreit magnetic resonance electrical impedance tomography mreit takes advantage of internal information to solve its nonlinear inverse problem of recovering a conductivity distribution inside an imaging object when we inject current into the imaging object there occurs a distribution of internal magnetic flux density mathbf b b x b y b z in mreit we utilize a magnetic resonance imaging scanner with its main magnetic field in the z direction to acquire b z data the harmonic b z algorithm was invented in 2001 to reconstruct cross sectional conductivity images from b z data sets subject to multiple injection currents utilizing internal b z data it overcomes the inherent ill posedness in electrical impedance tomography we can set up the inverse problem in mreit as a coefficient identification problem of finding sigma appearing in nabla cdot sigma nabla u 0 from acquired data of the z component of nabla times sigma nabla u the harmonic b z algorithm has shown an excellent perfor

mukai flops and deformations of symplectic resolutions we prove that two projective symplectic resolutions of open image in new window are connected by mukai flops in codimension 2 for a finite sub group g sp 2n it is also shown that two projective symplectic resolutions of open image in new window are deformation equivalent

a new average case analysis for completion time scheduling we present a new average case analysis for the problem of scheduling n jobs on m machines so that the sum of job completion times is minimized our goal is to use the concept of competitive ratio which is a typical worst case notion also within an average case analysis we show that the classic sept scheduling strategy with n worst case competitive ratio achieves an average of o 1 under several natural distributions among them the exponential distribution our analysis technique allows to also roughly estimate the probability distribution of the competitive ratio thus our result bridges the gap between worst case and average case performance guarantee

direct reconstructions of conductivities from boundary measurements the problem of reconstructing an unknown electric conductivity from boundary measurements has applications in medical imaging geophysics and nondestructive testing a nachman ann of math 2 143 1996 pp 71 96 proved global uniqueness for the two dimensional inverse conductivity problem using a constructive method of proof based on this proof siltanen mueller and isaacson inverse problems 16 2000 pp 681 699 presented a new numerical reconstruction method that solves the nonlinear problem directly without iteration the method was verified with nonnoisy rotationally symmetric examples in this paper the method is extended by introducing a new regularization scheme which is analyzed theoretically and tested on symmetric and nonsymmetric numerical examples containing computer simulated noise

the expected asymptotical ratio for preemptive stochastic online problem this paper considers the preemptive stochastic online scheduling problem on m uniform parallel machines with the objective to minimize the total weighted completion times where the processing times of jobs are independent random variables the actual processing time of each job is only learned upon its completion in online environment the existence and all parameters of each job including its weight release date and the distribution of processing time are unknown until its release date during the processing of each job it could be preempted and resumed later without extra cost for this problem a scheduling decision has to be made each time as the information of the processing time of the job is released step by step building on the gittins index we devise the semi preemptive policy for uniform machine problem sp policy with the assumption that all weights and processing times are bounded we prove the expected asymptotical ratio of sp is one which means e w jc jopt approaches to one as the number of jobs increases to infinity where c j denotes the completion time of job j j and opt the offline optimum value

norm estimates for k plane transforms and geometric inequalities abstract the article is devoted to remarkable interrelation between the norm estimates for k plane transforms in weighted and unweighted l p spaces and geometric integral inequalities for cross sections of measurable sets in r n a similar interrelation is studied for more general j plane to k plane transforms on affine grassmannians and their compact modifications the article contains a series of new integral geometric inequalities with sharp constants explicit equalities conjectures and open problems

the asymptotic performance ratio of an on line algorithm for uniform parallel machine scheduling with release dates jobs arriving over time must be non preemptively processed on one of m parallel machines each running at its own speed so as to minimize a weighted sum of the job completion times in this on line environment the processing requirement and weight of a job are not known before the job arrives the weighted shortest processing requirement wspr heuristic is a simple extension of the well known wspt heuristic which is optimal for the single machine problem without release dates according to wspr whenever a machine completes a job the next job assigned to it is the one with the least ratio of processing requirement to weight among all jobs available for processing at this point in time we analyze the performance of this heuristic and prove that its asymptotic competitive ratio is one for all instances with bounded job processing requirements and weights this implies that the wspr algorithm generates a solution whose relative error approaches zero as the number of jobs increases our proof does not require any probabilistic assumption on the job parameters and relies extensively on properties of optimal solutions to a single machine relaxation of the problem

the radon transform on sl 2 r so 2 r let g be sl 2 r g acts on the upper half plane u by the mobius transformation providing m with the riemannian metric structure along with the laplacian a we study the integral transform along each geodesic g acts on p the space of all geodesies in a natural way providing p with its invariant measure and its own laplacian p actually is a coset space of g therefore the above transform can be viewed as a map from a suitable function space on m to a suitable function space on p we prove a number of properties of this transform including the intertwining properties with its laplacians and its relation to the fourier transforms 1 introduction both the radon transform and the x ray transform on a symmetric space arise from the problem of reconstructing a function from its inte grals along certain paths for the euclidean case a problem can be stated in one of two ways suppose all the integrals of some function along all straight lines are known it is then possible to reconstruct from its line integrals for r2 the solution to this problem was the inversion of the original radon john transform the ability to invert this particular transform rested in a duality in integral geom etry between the points in r2 and the set of lines in r2 technically the inversion depends heavily on fourier analysis on r2 alternatively if the space in question is rn one might wish to reconstruct from its integrals over hyperplanes of dimen sion k solutions to both of these problems can be found in helgason 4 9 for the euclidean case we shall speak of the radon transform when we mean an integral over a fc plane where k n and of the x ray transform when we mean an integral over a straight line some references for the euclidean case include helgason 5 9 strichartz 17 radon 13 peters 12 solmon 15 16 and smith solmon and wagner 14 when one wishes to define the analogous transform on a general symmetric space several options become available the source of these options lies in the generalization of the notion of hyperplane in r2 for example there are two ways of thinking of an n l dimensional hyperplane one is as a totally geodesic n 1 submanifold the term totally geodesic means that a straight line tangent to a particular plane is actually contained in it thus one generalization of the radon transform which has been exploited is the non euclidean case of integrals over totally geodesic submanifolds of dimension k a theory including inversion formula has been worked out for the case where k is even in helgason 9 more can be found in gel fand et al 2 and lax and phillips 11

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

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

termination of flops for 4 folds the purpose of this note is to prove the termination of flops for 4 folds i e the second part of the so called flop conjecture in dimension 4 which asserts that there is no infinite sequence of flops with respect to a fixed reference divisor for projective 4 folds having only terminal singularities our idea for the proof stems from crossbreeding the ideas of kollar in ko3 where he proves the termination of flops for 3 folds with those of kawamata in kmm where he shows the termination of flips for 4 folds we should mention that the notion of difficulty due to shokurov si plays the most basic role in the ar

estimates for the x ray transform restricted to line complexes of kirillov type let there be given a piecewise continuous rectifiable curve r r n let g 1 n m 1 n be the usual grassmannian bundle in r n define an n dimensional submanifold m r n of m 1 n as the set of all copies of g 1 n along the curve following krillov we know that a nice function f x can be recovered from its x ray transform r 1 n f on m r n if and only if the curve intersects almost every affine hyperplane

distances between measures from 1 dimensional projections as implied by continuity of the inverse radon transform distances between measures on ir d are determined from distances between their 1 dimensional projections the method employed involves considering the 1 dimensional projections to be the radon transform of the measures crucial to the main theorem is a continuity result for the inverse radon transform focus is restricted to the prohorov dual bounded lipschitz and d k metrics which metrize weak convergence of probability measures these metrics are related to each other and to the sobolev norms the d k results extend from measures to generalized functions

dynamic scheduling of a multiclass fluid network a fluid network is a deterministic network model in which dynamic continuous flows are circulated and processed among a set of stations a fluid network often describes the asymptotic behavior of a stochastic queueing network via functional strong law of large numbers we study the dynamic scheduling of multiple classes of fluid traffic in such a network an algorithm is developed that systematically solves the dynamic scheduling problem by solving a sequence of linear programs it generates a policy in the form of dynamic capacity allocation at each station among all fluid classes that consists of a finite set of linear pieces over the entire time horizon in a single station or equivalently single server network this solution procedure recovers the priority index set that is optimal for the corresponding discrete queueing model generally known as klimov s problem

inversion of k plane transforms and applications in computer tomography the mathematics behind computerized tomography ct is based on the study of the parallel beam transform p and the divergent beam transform d both of these map a function f in mathbb r n into a function defined on the set of all lines in mathbb r n by integrating f along these lines the parallel and divergent k plane transforms are defined in a similar fashion by integration over k planes i e translates of k dimensional subspaces and are also denoted p and d respectively a related transform is the spherical k plane transform s which maps a function f on the sphere s n 1 into its integrals over k dimensional great circlesthis paper discusses the properties of the k plane transforms p d and s and their inverses emphasizing relationships and similarities between the operators and their relation to ct some new results are included most notable are more general conditions under which the inversion formulas hold

integral geometry on trees in the hyperbolic disc or more generally in real hyperbolic spaces we consider the horocyclic radon transform r and the geodesic radon transform x composition with their respective adjoint operators yields two convolution operators on the disc with respect to the hyperbolic measure we describe their convolution kernels in comparison with those of the corresponding operators on a homogeneous tree

cone beam reconstruction with sources on a curve an inversion procedure is developed for reconstructing a function of compact support in mathbb r 3 from its divergent beam x ray transform with sources on a curve the results apply to many curves not satisfying the conditions of tuy siam j appl math 43 1983 pp 546 552 for example planar circles of sufficiently large radius in some cases estimates in sobolev norms are established for the inversion operator

minimizing mean flow time with release time constraint abstract we consider the problem of preemptively scheduling a set of n independent tasks with release times on m identical processors with the objective of minimizing the mean flow time for one processor baker gives an o n log n time algorithm to find an optimal schedule lawler asks the question whether the problem can be shown to be solvable in polynomial time or shown to be np hard for m 2 in this paper we answer this question by showing that it is np hard for each fixed m 2

decompositions network flows and a precedence constrained single machine scheduling problem we present an in depth theoretical algorithmic and computational study of a linear programming lp relaxation to the precedence constrained single machine scheduling problem 1 prec s jw jc jto minimize a weighted sum of job completion times on the theoretical side we study the structure of tight parallel inequalities in thelp relaxation and show thatevery permutation schedule that is consistent with sidney s decomposition has total cost no more than twice the optimum on the algorithmic side we provide a parametric extension to sidney s decomposition and show that a finest decomposition can be obtained by essentially solving a parametric minimum cut problem finally we report results obtained by an algorithm based on these developments on randomly generated instances with up to 2 000 jobs