simultaneous minimization of total completion time and total deviation of job completion times abstract this paper addresses a single machine scheduling problem with the objective of minimizing a linear combination of total job completion times and total deviation of job completion times from a common due date the due date is assumed to be non restrictive i e sufficiently large to have no impact on the optimal sequence when the weights are job independent the problem is shown to have a polynomial time solution and the optimal schedule is fully characterized as a function of the different parameters when job dependent weights are assumed the problem is known to be np hard we introduce a pseudo polynomial dynamic programming algorithm indicating that this case is np hard in the ordinary sense the algorithm is shown experimentally to perform extremely well when tested on high multiplicity instances with up to 1000 jobs

vector coherent state constructions of u 3 symmetric tensors and their su 3 su 2 u 1 wigner coefficients generalized vector coherent state constructions of totally symmetric u 3 tensors are used to gain new expressions for the su 3 su 2 u 1 wigner coefficients for the coupling 1 1 20 3 3 these expressions show how the extremely simple formulas of le blanc and biedenharn involving a single 9 j coefficient arise as special cases of a general result that involves 12 j coefficients a simpler general result involving only 9 j coefficients and k normalization factors is derived in a way that can in principle be generalized to the generic coupling with multiplicity

clebsch gordan coefficients of su 3 with simple symmetry properties using an appropriate labelling operator constructed from representation generators su 3 clebsch gordan coefficients are introduced whose symmetry properties are similar to those of their su 2 counterparts an algebraic algorithm for computing the coefficients is presented

quantum k systems on generalise la notion classique d un systeme k a un systeme dynamique non commutatif on suppose une perte de memoire invariablement de 100 on donne des exemples de systemes k et on montre qu ils ne peuvent contenir aucun sous systeme quasi periodique

on the denominator function for canonical su 3 tensor operators the definition of a canonical unit su 3 tensor operator is given in terms of its characteristic null space as determined by group theoretic properties of the intertwining number this definition is shown to imply the canonical splitting conditions used in earlier work for the explicit and unique up to phases construction of all su 3 wcg coefficients wigner clebsch gordan using this construction an explicit su 3 invariant denominator function characterizing completely the canonically defined wcg coefficients is obtained it is shown that this denominator function squared is a product of linear factors which may be obtained explicitly from the characteristic null space times a ratio of polynomials these polynomials denoted gtq are defined over three shift parameters and three barycentric coordinates the properties of these polynomials hence of the corresponding invariant denominator function are developed in detail these include a derivation of their degree symmetries and zeros the

extended newton methods for multiobjective optimization majorizing function technique and convergence analysis we consider the extended newton method for approaching a pareto optimum of a multiobjective optimization problem establish quadratic convergence criteria and estimate a radius of convergence ball

dynamical entropy of c algebras and von neumann algebras the definition of the dynamical entropy is extended for automorphism groups ofc algebras as an example the dynamical entropy of the shift of a lattice algebra is studied and it is shown that in some cases it coincides with the entropy density

explicit canonical tensor operators and orthonormal coupling coefficients of su 3 the canonical unit su 3 tensor operators are constructed by means of the stretched coupling of the auxiliary maximal and minimal null space tensor operators with the renormalization factors expressed in terms of the denominator functions of biedenharn gustafson lohe louck and milne the matrix elements of the maximal null space tensor operators are expressed with the help of the modified projection operators of asherova and smirnov the self conjugate minimal null space tensor operators are expressed in terms of the group generators with the help of the weight lowering operator technique the corresponding extreme isoscalar factors of the clebsch gordan wigner coefficients are used as constructive elements of the explicit recursive expression for the general orthonormal isoscalar factors of su 3 with its considerable simplication for the boundary values of parameters the general isofactors are also expanded in the different ways in terms of their boundary values the new classes of the generaliz

earliness tardiness scheduling problems ii derivation of completion times about a restrictive common due date a companion paper part i considers the problem of minimizing the weighted earliness and tardiness of jobs scheduled on a single machine around a common due date d which is unrestrictively late this paper part ii considers the problem of minimizing the unweighted earliness and tardiness of jobs allowing the possibility that d is early enough to constrain the scheduling decision we describe several optimality conditions the recognition version of the problem is shown to be np complete in the ordinary sense confirming a well known conjecture moreover this complexity definition is precise since we describe a dynamic programming algorithm which runs in pseudopolynomial time this algorithm is also extremely efficient computationally providing an improvement over earlier procedures of almost two orders of magnitude in the size of instance that can be solved finally we describe a special case of the problem which is polynomially solvable

symmetry properties of matrix elements of canonical su 3 tensor operators the symmetries of the su 3 3 j symbols which are defined as symmetrized matrix elements of the canonical su 3 tensor operators are investigated the symmetries considered are those which in su 2 correspond to the interchange of columns of the 3 j symbol as well as the symmetry under conjugation it is found that for each tensor operator in a multiplicity set the matrix elements for a fixed operator pattern carry a one dimensional representation of the symmetric group s3

su3 isoscalar factors a summary of the properties of the wigner clebsch gordan coefficients and iso scalar factors for the group su3 in the su2 u1 decomposition is presented the outer degeneracy problem is discussed in detail with a proof of a conjecture braunschweig s which has been the basis of previous work on the su3 coupling coefficients recursion relations obeyed by the su3 isoscalar factors are produced along with an algorithm which allows numerical determination of the factors from the recursion relations the algorithm produces isoscalar factors which share all the symmetry properties under permutation of states and conjugation which are familiar from the su2 case the full set of symmetry properties for the su3 wigner clebsch gordan coefficients and isoscalar factors are displayed

implementation of the u n tensor operator calculus in a vector bargmann hilbert space using a vector coherent state framework and an associated vector bargmann hilbert space 1961 the structure of two classes of u n u n 1 unit projective operators is shown to be intimately related to 6 j and 9 j coefficients of the u n 1 subalgebra explicit verification of limit properties for these operators allows the unambiguous assignment of a set of canonical operator labels

minimizing total completion time and total deviation of job completion times from a restrictive due date we study a single machine scheduling problem with the objective of minimizing a linear combination of total job completion times and total deviation of job completion times from a common due date the due date is assumed to be restrictive i e it may be sufficiently small to have an impact on the optimal sequence when more weight is allocated to total job completion times the problem is shown to have a polynomial time solution when more weight is allocated to total completion time deviations from the due date the problem is np hard in the ordinary sense for the latter case we introduce an efficient dynamic programming algorithm which is shown numerically to perform well in all our tests

on biorthogonal and orthonormal clebsch gordan coefficients of su 3 analytical and algebraic approaches minimal biorthogonal systems of the clebsch gordan wigner coefficients of su 3 u 2 are discussed as well as the dual coupled bases the closed system of analytical expressions for the dual isofactors reduced wigner coefficients and the overlaps of coupled states is obtained with the help of analytical inversion symmetry the regge type symmetry of the overlaps and the boundary orthonormal isofactors orthogonalization coefficients is discovered the polynomial structure of the alternative complete algebraic systems of the orthonormal su 3 isofactors characterized by the null spaces symmetries and additional selection rules and obtained by means of the hecht or gram schmidt process is considered the realizations of the external missing label operators of the third and the fourth orders in the minimal coupled bases which lead to preferable algorithms to evaluate the orthonormal su 3 coupling coefficients satisfying different symmetry properties are presented with the help of the 6j coef

single machine scheduling with coefficient of variation minimization abstract the problem of scheduling n jobs on a single machine so as to minimize the coefficient of variation of the job completion times is considered in this paper some results on the characterization of an optimal schedule including its v shape are proved explicit optimal schedules for some small values of n are also given a heuristic method for solving the problem is proposed and its efficiency is investigated

a calculus for su 3 leading to an algebraic formula for the clebsch gordan coefficients we develop a simple computational tool for su 3 analogous to bargmann s calculus for su 2 crucial new inputs are i explicit representation of the gelfand zetlin basis in terms of polynomials in four variables and positive or negative integral powers of a fifth variable ii an auxiliary gaussian measure with respect to which the gelfand zetlin states are orthogonal but not normalized iii simple generating functions for generating all basis states and also all invariants as an illustration of our techniques an algebraic formula for the clebsch gordan coefficients is obtained for the first time this involves only gaussian integrations thus su 3 is made as accessible for computations as su 2 is

earliness tardiness scheduling problems i weighted deviation of completion times about a common due date this paper and its companion part ii concern the scheduling of jobs with cost penalties for both early and late completion in part i we consider the problem of minimizing the weighted sum of earliness and tardiness of jobs scheduled on a single processor around a common due date d we assume that d is not early enough to constrain the scheduling decision the weight of a job does not depend on whether the job is early or late but weights may vary between jobs we prove that the recognition version of this problem is np complete in the ordinary sense we describe optimality conditions and present a computationally efficient dynamic programming algorithm when the weights are bounded by a polynomial function of the number of jobs a fully polynomial approximation scheme is given we also describe four special cases for which the problem is polynomially solvable part ii provides similar results for the unweighted version of this problem where d is arbitrary

new perspective on the u n wigner racah calculus ii elementary reduced wigner coefficients for u n for pt i see ibid vol 20 no 9 p 2241 50 1987 exploiting the powerful formalism of the vector coherent state theory expounded in part i the group theoretical content of the complementarity principle and a novel interpretation of the operator pattern of biedenharn and louck the authors rederive in a simple fashion all u n u n 1 reduced wigner coefficients for the elementary u n tensors

a resolution of the su 3 outer multiplicity problem and computation of wigner coefficients for su 3 a simple algorithm is given for the resolution of the su 3 multiplicity problem and the computation of su 3 wigner coefficients using a complete set of u 2 x u 3 bargmann tensors classified by operator patterns null space properties of these tensors are easily derived their structure is such that a direct one to one correspondence is shown to exist between the bargmann tensors and the terms in the clebsch gordan series for su 3 as derived by o reilly 1982 finally the resolution presented herein is shown to be concordant with an alternative resolution advocated long ago by hecht

minimizing mean squared deviation of completion times about a common due date this paper addresses a nonpreemptive single machine scheduling problem where all jobs have a common due date and have zero ready time the scheduling objective is to minimize mean squared deviation msd of job completion times about the due date this nonregular measure of performance is appropriate when earliness and tardiness are both penalized and when large deviations of completion time from the due date are undesirable a special case of the msd problem referred to as the unconstrained msd problem is shown to be equivalent to the completion time variance problem ctv studied by merten and muller merten a g m e muller 1972 variance minimization in single machine sequencing problems management sci 18september 518 528 and schrage schrage l 1975 minimizing the time in system variance for a finite jobset management sci 21may 540 543 strong results for this latter problem are combined with several new propositions to develop a reasonably efficient procedure for solving the unconstrained msd problem this enables us to improve the existing procedures for the ctv problem we also propose a branching procedure for the constrained msd problem and present computational results

on the minimization of completion time variance with a bicriteria extension we discuss a single machine scheduling problem where the objective is to minimize the variance of job completion times to date the problem has not been solved in polynomial time this paper presents a dynamic programming algorithm that is pseudopolynomial in complexity we also propose a fully polynomial approximation scheme and derive a lower bound that is useful in its implementation furthermore we show that the dynamic programming solution is easy to extend to a bicriteria version of the problem in which it is desired to simultaneously minimize the mean completion time

symmetry properties of s classified su 3 3j 6j and 9j symbols starting from the s classified su 3 clebsch gordan coefficients introduced previously pluhar et al 1985 the s classified su 3 3j 6j and 9j symbols are constructed and examined they satisfy simple symmetry relations similar to those valid for the su 2 case

the variational principle for a class of asymptotically abelian c algebras let a be a c dynamical system we introduce the notion of pressure p h of the automorphismat a self adjoint operatorh a then we consider the class of af systems satisfying the following condition there exists a dense invariant subalgebra a of a such that for all pairs a b a the c algebra they generate is finite dimensional and there is p p a b n such that j a b 0 for j p for systems in this class we prove the variational principle i e show that p h is the supremum of the quantities h h where h is the connes narnhofer thirring dynamical entropy ofwith respect to the invariant state if h a and p h is finite we show that any state on which the supremum is attained is a kms state with respect to a one parameter automorphism group naturally associated with h in particular voiculescu s topological entropy is equal to the supremum of h and any state of finite maximal entropy is a trace

scheduling about a given common due date to minimize mean squared deviation of completion times abstract in this paper the problem of minimizing the mean squared deviation msdp of job completion times about a given common due date in n job single machine scheduling is considered the release times for all jobs are assumed to be zero no polynomial time algorithms have been found to identify an optimal solution to msdp for the tightly restricted version of msdp no even pseudo polynomial time algorithms have been found to identify an optimal solution we present some dominance properties for msdp based on which a pseudo polynomial dynamic programming algorithm which is an extension of the dynamic programming algorithm proposed by lee danusaputro and lin is presented to optimally solve the tightly restricted version of msdp computational results show that the proposed dynamic programming algorithm can optimally solve tightly restricted instances of msdp with up to 100 jobs in less than two seconds

non bernoullian quantum k systems we construct an uncountable family of pairwise non conjugate non bernoullian k systems of type iii1 with the same finite cnt entropy we also investigate clustering properties of multiple channel entropies for strong asymptotically abelian systems of type ii and iii we prove that a wide enough class of systems has the k property in particular such systems as the space translations of a one dimensional quantum lattice with the gibbs states of araki the space translations of the ccr algebra and the even part of the car algebra with the quasi free states of park and shin non commutative markov shifts in the accardi sense are entropic k systems

gibbs states for af algebras we consider a special class of c systems containing asymptotically abelian binary shifts and shifts of temperley lieb algebras we study gibbs states for these systems corresponding to potentials with finite range interaction and obtain the same results as the well known araki s results for a one dimensional quantum lattice in particular it is proved that a gibbs state in the infinite volume is a translation invariant kms state having the exponential uniform clustering property entropic properties of the gibbs states are also discussed this allows us in particular to construct new examples of quantum k systems

minimizing weighted mean absolute deviation of job completion times from their weighted mean we address a single machine scheduling problem where the objective is to minimize the weighted mean absolute deviation of job completion times from their weighted mean this problem and its precursors aim to achieve the maximum admissible level of service equity it has been shown earlier that the unweighted version of this problem is np hard in the ordinary sense for that version a pseudo polynomial time dynamic program and a 2 approximate algorithm are available however not much except for an important solution property exists for the weighted version in this paper we establish the relationship between the optimal solution to the weighted problem and a related one in which the deviations are measured from the weighted median rather than the mean of the job completion times this generalizes the 2 approximation result mentioned above we proceed to give a pseudo polynomial time dynamic program establishing the ordinary np hardness of the problem in general we then present a fully polynomial time approximation scheme as well finally we report the findings from a limited computational study on the heuristic solution of the general problem our results specialize easily to the unweighted case they also lead to an approximation of the set of schedules that are efficient with respect to both the weighted mean absolute deviation and the weighted mean completion time

on the structure of tensor operators in su3 a global algebraic formulation of su3 tensor operator structure is achieved a single irreducible unitary representation irrep v ofso 6 2 is constructed which contains every su3 irrep precisely once an algebra of polynomial differential operatorsa acting onv is given the algebraa is shown to consist of linear combinations of all su3 tensor operators with polynomial invariant operators as coefficients by carrying out an analysis ofa the multiplicity problem for su3 tensor operators is resolved

explicit orthonormal clebsch gordan coefficients of su 3 the construction of the explicit algebraic polynomial expressions for the nonmultiplicity free orthonormal clebsch gordan wigner coefficients of su 3 u 2 is completed in the case of the paracanonical coupling scheme related with the explicit minimal biorthogonal systems by means of the hecht or gram schmidt process the direct and inverse orthogonalization coefficients the first of them being equivalent to the boundary orthonormal isofactors are expressed up to explicitly given multiplicative factors in terms of the numerator and denominator polynomials related with the auxiliary a function of louck biedenharn and lohe that appears as a fragment of the denominator g functions of canonical su 3 tensor operators

large deviations in quantum spin chains we show the full large deviation principle for kms states and c finitely correlated states on a quantum spin chain we cover general local observables our main tool is ruelle s transfer operator method

deterministic and random single machine sequencing with variance minimization we discuss the problem of scheduling several jobs on a single machine so as to minimize the variance of the jobs completion times by exploiting the well known pooled variance formula we obtain a general formula for the change in variance due to the interchange of two specified jobs whose positions have not been fixed in the sequence this result permits us to obtain explicit optimal sequences for problems with 6 or 7 jobs and to show that schrage s conjecture on positioning the job with the third largest processing time is true for problems with up to 18 jobs we then develop a heuristic procedure based on a variance interchange formula and compare it with other known heuristic methods we also consider a problem never before examined namely the case when processing times are not known deterministically but are random variables we obtain some general results under certain assumptions on the parameters of the distributions of processing times and extend to the random situation many results applicable to the deterministic case finally we give some results for the exponential distribution of processing times

complexity of single machine multi criteria scheduling problems abstract we examine the complexity of scheduling problems when more than one measure of performance is appropriate the criteria we study are maximal tardiness flowtime number of tardy jobs tardiness and the weighted counterparts of the last three measures the machine environment is restricted to a single machine complexity results are given for secondary criterion bicriteria and weighted criteria approaches for all combinations of measures of the problems examined only six remain open

complementary group resolution of the su n outer multiplicity problem i the littlewood rules and a complementary u 2n 2 group structure a complementary group to su n is found that realizes all features of the littlewood rules for kronecker products of su n representations this is accomplished by considering a state of su n to be a special gel fand state of the complementary group u 2n 2 with labels of the latter used to distinguish multiple occurrences of irreducible representations of su n irreps in the su n su n su n decomposition that is obtained from the littlewood rules furthermore this realization also helps us to determine su n su n 1 u 1 reduced wigner coefficients rwcs frequently called isoscalar factors and clebsch gordan coefficients cgcs or full nonreduced wigner coefficients of su n using algebraic or numeric methods in either the canonical or a noncanonical basis new explicit formulas for the su 3 and su 4 multiplicities are obtained by using this technique

a closed formula for the product of irreducible representations of su 3 we determine a closed formula in terms of p q r and s for the decomposition of the product p q r s of finite dimensional irreducible representations of su 3 we also determine in terms of p q r s m and n necessary and sufficient conditions that a term m n appears in this decomposition and its multiplicity

orthogonalization of the projected states and isofactors the explicit direct and inverse orthogonalization coefficients ocs are derived for the projected basis states of the two parametric covariant irreducible representations of including elliott s n 3 case and draayer with the alternative choices of the gram schmidt processes began respectively from the lowest vergados or highest absolute value of the intrinsic parameter k the different direct and inverse ocs the latter equivalent to the definite boundary isofactors or the boundary resubducing coefficients are found to be connected by the definite analytical continuation contraction procedures with mutually related ocs of the biorthogonal states and the boundary orthonormal pseudocanonical and paracanonical isofactors of the orthogonalization coefficients considered are presented as definite algebraic polynomial structures under the square root with the linear factors and numerator denominator polynomials in terms of the partition dependent functions of biedenharn and louck

the missing link operators for labelling multiplicity in the clebsch gordan series it is indicated that there are invariant operators on the direct product space of irreducible representations of lie groups which distinguish between repeating representations they complete the labelling of the clebsch gordan series

note a note on the minimization of mean squared deviation of completion times about a common due date this paper points out a conceptual flaw present in a recent publication it shows how the methodology can be modified to correct this flaw it also presents improved bounds to determine the nature of a given problem

on the k property of quantized arnold cat maps we prove that some quantized arnold cat maps are entropic k systems this result was formulated by h narnhofer j math phys 33 1502 1510 1992 but the fact that the optimal decomposition for the multi channel entropy constructed there is not strictly local was not appropriately taken care of we propose a strictly local decomposition based on a construction of voiculescu