minimizing total tardiness on one machine is np hard the problem of minimizing the total tardiness for a set of independent jobs on one machine is considered lawler has given a pseudo polynomial time algorithm to solve this problem in spite of extensive research efforts for more than a decade the question of whether it can be solved in polynomial time or it is np hard in the ordinary sense remained open in this paper the problem is shown to be np hard in the ordinary sense

incomplete inverse spectral problems for dirac bessel operators we investigate the incomplete inverse spectral problems for dirac bessel operators defined on 0 1 we show that when the potential is known on the subinterval a 1 0 1 the potential on the whole interval can be uniquely determined in terms of appropriate partial information on the spectral data including eigenvalues and norming constants moreover these results can be applied to solve the incomplete inverse spectral problems for a class of bessel operators we investigate the incomplete inverse spectral problems for dirac bessel operators defined on 0 1 we show that when the potential is known on the subinterval a 1 0 1 the potential on the whole interval can be uniquely determined in terms of appropriate partial information on the spectral data including eigenvalues and norming constants moreover these results can be applied to solve the incomplete inverse spectral problems for a class of bessel operators

uqosp 2 2 lattice models in this paper i construct lattice models with an underlying uqosp 2 2 superalgebra symmetry i find new solutions to the graded yang baxter equation these trigonometric r matrices depend on three continuous parameters the spectral parameter the deformation parameter q and the u 1 parameter b of the superalgebra it must be emphasized that the parameter q is generic and the parameter b does not correspond to the nilpotency parameter of the rational limits are given they also depend on the u 1 parameter and this dependence cannot be rescaled away i give the bethe ansatz solution of the lattice models built from some of these r matrices while for other matrices due to the particular nature of the representation theory of osp 2 2 i conjecture the result the parameter b appears as a continuous generalized spin finally i briefly discuss the problem of finding the ground state of these models

comparing efficiencies of genetic crossover operators for one machine total weighted tardiness problem in this study the well known one machine problem with the performance criterion of minimizing total weighted tardiness is considered this problem is known to be np hard and consists of one machine and n independent jobs each of these jobs has a distinct integer processing time a distinct integer weighting factor and a distinct integer due date the purpose of this problem is to find a sequence of these jobs minimizing the sum of the weighted tardiness using benchmarking problems this study compares performances of eleven genetic crossover operators which have been widely used to solve other types of hard scheduling problems

the quantum group structure of 2d gravity and minimal models i on the unit circle an infinite family of chiral operators is constructed whose exchange algebra is given by the universalr matrix of the quantum groupsl 2 q this establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group the method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofsl 2 q the formulae so derived which generalize an earlier particular case worked out by babelon are remarkably compact and may be entirely written in terms of q deformed factorials and binomial coefficients

inverse spectral theory for a singular sturm liouville operator on 0 1 xer3 acting on functions in p 1x1 1 vanishing on 1x1 1 we set x 1x1 and assume q x is a real valued function in l 0 11 the closure h of i is a self adjoint operator with compact resolvent hence h has pure point spectrum without finite accumulation points moreover using the mapping u x xu x and expansion in spherical harmonics one shows that h is unitarily equivalent to a direct sum of sturm liouville operators acting on l o 11 cf 11 the first operator in this sum is

multiparameter deformations of the universal enveloping algebras of the simple lie algebras al for all l 2 and the yang baxter equation a 1 1 2l l 1 parameter deformation of the universal enveloping algebra of the simple lie algebra al sl l 1 c is derived for all l 2 starting from a certain multiparameter matrix that satisfies the quantum yang baxter equation it is shown that this hopf algebra has the same product relations and antipode as the standard one parameter deformation uq sl l 1 c but has a different coproduct it is also shown that for all l 2 there exists a 1 2l l 1 parameter hopf algebra whose product relations are merely the commutation relations of al itself but whose coproduct is different from the usual one for the universal enveloping algebra of al

classification of quantum groups and lie bialgebra structures on sl n f relations with brauer group given an arbitrary field f of characteristic 0 we study lie bialgebra structures on sl n f based on the description of the corresponding classical double for any lie bialgebra structure 5 the classical double d sl n f delta is isomorphic to sl n f circle times f a where a is either f epsilon with epsilon 2 0 or f circle plus f or a quadratic field extension of f in the first case the classification leads to quasi frobenius lie subalgebras of sl n f in the second and third cases a belavin drinfeld cohomology can be introduced which enables one to classify lie bialgebras on sl n f up to gauge equivalence the belavin drinfeld untwisted and twisted cohomology sets associated to an r matrix are computed for the cremmer gervais r matrix in sl 3 we also construct a natural map of sets between the total belavin drinfeld twisted cohomology set and the brauer group of the field f

on dynamical r matrices obtained from dirac reduction and their generalizations to affine lie algebras according to etingof and varchenko the classical dynamical yang baxter equation is a guarantee for the consistency of the poisson bracket on certain poisson lie groupoids here it is noticed that dirac reductions of these poisson manifolds give rise to a mapping from dynamical r matrices on a pair to those on another pair where is a chain of lie algebras for which admits a reductive decomposition as several known dynamical r matrices appear naturally in this setting and its application provides new r matrices too in particular we exhibit a family of r matrices for which the dynamical variable lies in the grade zero subalgebra of an extended affine lie algebra obtained from a twisted loop algebra based on an arbitrary finite dimensional self dual lie algebra

a branch and bound algorithm for the total weighted tardiness problem this paper presents a new branch and bound algorithm for the single machine total weighted tardiness problem it obtains lower bounds using a lagrangian relaxation approach with subproblems that are total weighted completion time problems the well known subgradient optimization technique is replaced by a multiplier adjustment method that leads to an extremely fast bound calculation the method incorporates various devices for checking dynamic programming dominance in the search tree extensive computational results for problems with up to 50 jobs show the superiority of the algorithm over existing methods

solution of the inverse spectral problem for an impedance with integrable derivative part ii the inverse eigenvalue problem for a sturm liouville equation in impedance form with dirichlet boundary conditions on a unit interval is solved the solution of this nonlinear problem requires the investigation of a combined map from the logarithmic derivative of the impedance assumed to be square integrable to two sequences of spectral data the first is the sequence of dirichlet eigenvalues shown to be locally bounded with square roots differing from the sequence of integral multiples of by a square summable sequence the second sequence has as first term the mean of the logarithmic derivative of the impedance each term in the remainder of the sequence is the logarithm of the product of the impedance and the derivative of an eigenfunction evaluated at an endpoint it is shown that this second sequence is a locally bounded square summable sequence the combined map is a real analytic isomorphism a uniqueness result follows from the asymptotics and analyticity shown in the first paper and properties of riesz bases the existence is shown by giving a closed form of the solution it is also shown that the same map solves the inverse sturm liouville problem in impedance form with neumann boundary conditions 1993 john wiley sons inc

on the singular weyl titchmarsh function of perturbed spherical schrodinger operators we investigate the singular weyl titchmarsh m function of perturbed spherical schrodinger operators also known as bessel operators under the assumption that the perturbation q x satisfies xq x l1 0 1 we show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter based on this we show that the singular m function belongs to the generalized nevanlinna class and connect our results with the theory of super singular perturbations

the single machine total tardiness scheduling problem review and extensions abstract we review the latest theoretical developments for the single machine total tardiness 1 t scheduling problem and propose extensions to some of them we also review and in some cases extend exact algorithms fully polynomial time approximation schemes heuristic algorithms special cases and generalizations of the 1 t problem our findings indicate that the 1 t problem continues to attract significant research interest from both a theoretical and a practical perspective even though the problem is ordinary np hard the current state of the art algorithms are capable of solving problems with up to 500 jobs

reconstruction of radial dirac and schrodinger operators from two spectra abstract we solve the inverse problem of reconstructing radial dirac and schrodinger operators acting in the unit ball of r 3 from two spectra of their one dimensional parts corresponding to a fixed nonzero angular momentum we give a complete description of the spectral data prove existence and uniqueness of solutions to the inverse problem and present the reconstruction algorithm

reconstruction of a radially symmetric potential from two spectral sequences we consider the problem of determining a radially symmetric potential in the three dimensional schrodinger equation from eigenvalues associated with two different angular momentum quantum numbers this leads to a singular eigenvalue problem for which there are no known general uniqueness results for the most reasonable conjectures we are able to give strong evidence for uniqueness in some cases and we discuss a computational solution method together with some supporting analysis one application we have in mind is the determination of certain physical parameters in the standard model of the sun constructed from eigenvalue data

on the construction of unitary quantum group differential calculus we develop a construction of the unitary type anti involution for the quantized differential calculus over in the case to this end we consider a joint associative algebra of quantized functions differential forms and lie derivatives over which is bicovariant with respect to coactions we define a specific non central spectral extension of this algebra by the spectral variables of three matrices of the algebra generators in the spectrally expended algebra we construct a three parametric family of its inner automorphisms these automorphisms are used for the construction of the unitary anti involution for the spectrally extended calculus over

regular singular sturm liouville operators and their zeta determinants abstract we consider sturm liouville operators on the line segment 0 1 with general regular singular potentials and separated boundary conditions we establish existence and a formula for the associated zeta determinant in terms of the wronski determinant of a fundamental system of solutions adapted to the boundary conditions this generalizes the earlier work of the first author treating general regular singular potentials but only the dirichlet boundary conditions at the singular end and the recent results by kirsten loya park for general separated boundary conditions but only special regular singular potentials

a tabu search algorithm for the single machine total weighted tardiness problem abstract in this study a tabu search ts approach to the single machine total weighted tardiness problem smtwt is presented the problem consists of a set of independent jobs with distinct processing times weights and due dates to be scheduled on a single machine to minimize total weighted tardiness the theoretical foundation of single machine scheduling with due date related objectives reveal that the problem is np hard rendering it a challenging area for meta heuristic approaches this paper presents a totally deterministic ts algorithm with a hybrid neighborhood and dynamic tenure structure and investigates the strength of several candidate list strategies based on problem specific characteristics in increasing the efficiency of the search the proposed ts approach yields very high quality results for a set of benchmark problems obtained from the literature

classical flag varieties for quantum groups the standard quantum sl n c abstract we suggest a possible programme to associate geometric flag like data to an arbitrary simple quantum group in the spirit of the noncommutative al gebraic geometry developed by artin tate and van den bergh we then carry out this programme for the standard quantum sl n of drinfel d and jimbo where the varieties involved are certain t stable subvarieties of the ordinary flag variety

inverse spectral problems for sturm liouville operators in impedance form abstract the spectral properties of sturm liouville operators with an impedance in w p 1 0 1 for some p 1 are studied in particular a complete solution of the inverse spectral problem is provided

spectral asymptotics for canonical systems based on continuity properties of the de branges correspondence we develop a new approach to study the high energy behavior of weyl titchmarsh and spectral functions of 2 2 first order canonical systems our results improve several classical results and solve open problems posed by previous authors furthermore they are applied to radial dirac and radial schrodinger operators as well as to krein strings and generalized indefinite strings

approximation algorithms for minimizing the total weighted tardiness on a single machine given a single machine and a set of jobs with due dates the classical np hard problem of scheduling to minimize total tardiness is a well understood one lawler gave a fully polynomial time approximation scheme fptas for it some 20 years ago if the jobs have positive weights the problem of minimizing total weighted tardiness seems to be considerably more intricate it in this paper we give some of the first approximation algorithms for it we examine first the weighted problem with a fixed number of due dates and we design a pseudopolynomial algorithm for it we show how to transform the pseudopolynomial algorithm to an fptas for the case where the weights are polynomially bounded for the case with an arbitrary number of due dates and polynomially bounded processing times we provide a quasipolynomial algorithm which produces a schedule whose value has an additive error proportional to the weighted sum of the due dates we also investigate the performance of algorithms for minimizing the related total weighted late work objective

the inverse spectral problem for radial schrodinger operators on 0 1 we consider an inverse spectral problem for a class of singular sturm liouville operators on the unit interval with explicit singularity a a 1 x2 a n related to the schrodinger operator with a radially symmetric potential the purpose of this paper is the global parametrization of potentials by the spectral data a and some norming constants a for a 0 or 1 a a is already known to be a global coordinate system on lr2 0 1 using some transformation operators we show that this result holds for any non negative integer a moreover we give a description of the isospectral sets

weighted tardiness scheduling on parallel machines with proportional weights in this paper we address the problem of scheduling a number of jobs on a bank of parallel machines to minimize the total weighted tardiness under the assumption that the weight of each job is proportional to its processing time the version of the problem that has general weights has been shown to be strongly np complete we prove this version of the problem to be np complete and give a pseudopolynomial time algorithm for solving it we study a family of simple sequencing rules in which the jobs are sequenced in increasing order of i di pi where di is the due date of job i pi its processing time wi its weight and 0 1 this family of sequencing rules generalizes the earliest due date sequencing rule we obtain bounds on the ratio c c iwipi where c and c are the costs of the heuristic and optimal schedules respectively the denominator is the cost of having each job be late by its own processing time it is intended to measure what is or is not a large deviation from optimali

explicit quantization of dynamical r matrices for finite dimensional semisimple lie algebras 1 1 classical r matrices in the early eighties belavin and drinfeld bd classified nonskewsymmetric classical r matrices for simple lie algebras it turned out that such r matrices up to isomorphism and twisting by elements from the exterior square of the cartan subalgebra are classified by combinatorial objects which are now called belavin drinfeld triples by definition a belavin drinfeld triple for a simple lie algebra g is a triple 1 2 t where 1 2 are subsets of the dynkin diagram of g and t 1 2 is an isomorphism which preserves the inner product and satisfies the nilpotency condition if 1 then there exists k such that t k 1 1 but t 1 the r matrix corresponding to such a triple is given by a certain explicit formula these results generalize in a straightforward way to semisimple lie algebras in s the third author generalized the work of belavin and drinfeld and classified classical nonskewsymmetric dynamical r matrices for simple lie algebras it turns out that they have an even simpler classification up to gauge transformations they are classified by generalized belavin drinfeld triples which are defined as the usual belavin drinfeld triples but without any nilpotency condition the dynamical rmatrix corresponding to such a triple is given by a certain explicit formula as before these results can be generalized to semisimple lie algebras

t b q q analog of model space and the clebsch gordan coefficients generating matrices we study relations between the deformed cotangent bundle t b q for the borel subgroup b of a given simple lie group g the quantum lie algebra jq associated with the corresponding quantum group gq and the matrices generating clebsch gordan coefficients for jq we reveal the connection of these objects to quantum analog of the model space m and q tensor operators

on some lie bialgebra structures on polynomial algebras and their quantization we study classical twists of lie bialgebra structures on the polynomial current algebra mathfrak g u where mathfrak g is a simple complex finite dimensional lie algebra we focus on the structures induced by the so called quasi trigonometric solutions of the classical yang baxter equation it turns out that quasi trigonometric r matrices fall into classes labelled by the vertices of the extended dynkin diagram of mathfrak g we give the complete classification of quasi trigonometric r matrices belonging to multiplicity free simple roots which have coefficient 1 in the decomposition of the maximal root we quantize solutions corresponding to the first root of mathfrak sl n

classical a n w geometry by analyzing theextrinsic geometry of two dimensional surfaces chirally embedded incp n thecp n w surface 1 we give exact treatments in various aspects of the classical w geometry in the conformal gauge first the basis of tangent and normal vectors are defined at regular points of the surface such that their infinitesimal displacements are given by connections which coincide with the vector potentials of the conformal a n toda lax pair since the latter is known to be intrinsically related with the w symmetries this gives the geometrical meaning of thea n w algebra second w surfaces are put in one to one correspondence with solutions of the conformally reduced wznw model which is such that the toda fields give the cartan part in the gauss decomposition of its solutions third the additional variables of the toda hierarchy are used as coordinates ofcp n this allows us to show that w transformations may be extended as particular diffeomorphisms of this target space higher dimensional generalizations of the wznw equations are derived and related with the zakharov shabat equations of the toda hierarchy fourth singular points are studied from a global viewpoint using our earlier observation 1 that w surfaces may be regarded as instantons the global indices of the w geometry which are written in terms of the toda fields are shown to be the instanton numbers for associated mappings of w surfaces into the grassmannians the relation with the singularities of w surface is derived by combining the toda equations with the gauss bonnet theorem

single machine scheduling to minimize total weighted tardiness abstract the problem of minimizing the total weighted tardiness in single machine scheduling is a well known strongly np hard problem in this paper we present an o n 2 time approximation algorithm for the problem where n is the number of jobs we further investigate different sub models of the problem and obtain interesting properties and results we begin with the model where the job due dates are affine linear functions of their processing times and the job tardiness weights are proportional to their processing times for this model we classify the np hardness of the problem with respect to different affine linear functions and provide an o n log n algorithm for each of the polynomially solvable problems the second model we investigate is one where all job due dates have equal slacks namely the slk due date model results for this model include the problem is np hard in the ordinary sense a pseudo polynomial algorithm with time complexity o n 2 p is derived where p is the sum of all of the processing times and a fully polynomial time approximation scheme is constructed

the quantum group structure associated with non linearly extended virasoro algebras recently an infinite family of chiral virasoro vertex operators whose exchange algebra is given by the universalr matrix ofsl 2 q has been constructed in the present paper the case of non linearly w extended virasoro symmetries related to the algebrasan n 1 is considered along the same line contrary to the previous case a1 the standardr matrix ofsl n 1 q does not come out and a different solution of yang and baxter s equations is derived the associated quantum group structure is displayed

the non dynamical r matrices of the degenerate calogero moser models a complete description of the non dynamical r matrices of the degenerate calogero moser models based on gln is presented first the most general momentum independent r matrices are given for the standard lax representation of these systems and those r matrices whose coordinate dependence can be gauged away are selected then the constant r matrices resulting from the gauge transformation are determined and are related to well known r matrices in the hyperbolic trigonometric case a non dynamical r matrix equivalent to a real imaginary multiple of the cremmer gervais classical r matrix is found in the rational case the constant r matrix corresponds to the antisymmetric solution of the classical yang baxter equation associated with the frobenius subalgebra of gln consisting of the matrices with vanishing last row these claims are consistent with previous results of hasegawa and others which imply that belavin s elliptic r matrix and its degenerations appear in the calogero moser models the advantages of our analysis are that it is elementary and also clarifies the extent to which the constant r matrix is unique in the degenerate cases

hecke algebraic properties of dynamical r matrices application to related quantum matrix algebras the quantum dynamical yang baxter or gervais neveu felder equation defines an r matrix r p where p stands for a set of mutually commuting variables a family of sl n type solutions of this equation provides a new realization of the hecke algebra we define quantum antisymmetrizers introduce the notion of quantum determinant and compute the inverse quantum matrix for matrix algebras of the type r p a1a2 a1a2r it is pointed out that such a quantum matrix algebra arises in the operator realization of the chiral zero modes of the wznw model

numerical solution of the inverse spectral problem for bessel operators we consider some inverse spectral problems associated with the singular sturm liouville equation u q x 1 x 2 u lu0

an infinite product formula for uq sl 2 dynamical coboundary element we give a short summary of results and conjectures in the theory of dynamical quantum group related to the dynamical coboundary equation also known as irf vertex transform babelon has shown that the dynamical twist f x of uq sl 2 is a dynamical coboundary m x i e f x m 1 xq h 2 m 2 x a m x we give a new formula for this element m x as an infinite product and give a new proof of the coboundary relation our proof involves the quantum weyl group element giving a possible hint for the generalization to higher rank case

on two dimensional supergravity and the quantum super mobius group we describe the quantum group structure of two dimensionaln 1 supergravity in the conformal gauge the operator algebra of this super liouville theory is shown to correspond to a quantum deformation of the super mobius group which provides a new solution of yang and baxter s equation this structure is used to investigate the strong coupling regime of the theory 1 d 9 ford 5 a unitary truncation theorem similar to the one of the bosonic case is derived for the fundamental representation

a complete spectral characterization of the double commutation method abstract we provide a complete spectral characterization of the double commutation method which introduces eigenvalues into arbitrary spectral gaps of a given background schrodinger operator in l 2 r respectively in l 2 0 our main result proves unitary equivalence of the doubly commuted operator restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues with the original background operator moreover we explicitly compute the matrix spectral function of the doubly commuted operator in terms of the background spectral function

inverse spectral problems for bessel operators abstract we study the inverse spectral problem for a class of bessel operators given in l 2 0 1 by the differential expression d d x x v d d x x v with n and a real valued function v l p 0 1 p 1 subject to various boundary conditions we describe completely the spectral data of these operators i e the spectra and corresponding norming constants and give the algorithm of reconstruction of v from the spectral data

analytical methods for recovering coefficients in differential equations from spectral data in this paper analytical methods for recovering coefficients from spectral data are discussed properties of these methods are given and mathematical models for which these methods might apply are presented

quantum dynamical coboundary equation for finite dimensional simple lie algebras abstract for a finite dimensional simple lie algebra g the standard universal solution r x u q g 2 of the quantum dynamical yang baxter equation quantizes the standard trigonometric solution of the classical dynamical yang baxter equation it can be built from the standard r matrix and from the solution f x u q g 2 of the quantum dynamical cocycle equation as r x f 21 1 x r f 12 x f x can be computed explicitly as an infinite product through the use of an auxiliary linear equation the abrr equation inspired by explicit results in the fundamental representation it has been conjectured that in the case where g sl n 1 n 1 only there could exist an element m x u q sl n 1 such that the dynamical gauge transform r j of r x by m x r j m 1 x 1 m 2 x q h 1 1 r x m 1 x q h 2 m 2 x does not depend on x and is a universal solution of the quantum yang baxter equation in the fundamental representation r j corresponds to the standard solution r for n 1 and to cremmer gervais s one r 12 j j 21 1 r 12 j 12 for n 1 for consistency m x should therefore satisfy the quantum dynamical coboundary equation i e f x m x j m 2 x 1 m 1 x q h 2 1 in which j u q sl n 1 2 is the universal cocycle associated to cremmer gervais s solution the aim of this article is to prove this conjecture and to study the properties of the solutions of the quantum dynamical coboundary equation in particular by introducing new basic algebraic objects which are the building blocks of the gauss decomposition of m x we construct m x in u q sl n 1 as an explicit infinite product which converges in every finite dimensional representation we emphasize the relations between these basic objects and some non standard loop algebras and exhibit relations with the dynamical quantum weyl group

a fully polynomial approximation scheme for the single machine weighted total tardiness problem with a common due date we develop a fully polynomial time approximation scheme fptas for minimizing the weighted total tardiness on a single machine provided that all due dates are equal the fptas is obtained by converting an especially designed pseudopolynomial dynamic programming algorithm

quantum groups and deformation quantization explicit approaches and implicit aspects deformation quantization which gives a development of quantum mechanics independent of the operator algebra formulation and quantum groups which arose from the inverse scattering method and a study of yang baxter equations share a common idea abstracted earlier in algebraic deformation theory that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations i e quantizations in deformation quantization the algebraic object is the algebra of observables functions on symplectic phase space whose infinitesimal deformation is the poisson bracket and global deformation a star product in quantum groups it is a hopf algebra generally either of functions on a lie group or often its dual in the topological vector space sense as we briefly explain a completed universal enveloping algebra of a lie algebra with for infinitesimal a matrix satisfying the modified classical yang baxter equation mcybe frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract one success here comes from universal deformation formulas udfs expressions built from a lie algebra which deform any algebra on which the lie algebra operates as derivations the most famous of these is the moyal product a special case of a class in which the lie algebra is abelian another comes from recognition that the belavin drinfel d solutions to the mcybe are in fact infinitesimal deformations for which in the case of the special linear groups it is possible to give explicit formulas for the corresponding quantum yang baxter equations this review paper discusses necessarily in brief these and related topics including twisting as a form of udf and finding formulas for preferred deformations of hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation

twisted yang baxter equations for linear quantum super groups we consider the modified or twisted yang baxter equations for the groups and supergroups the general solutions for these equations are presented in the case of the linear quantum super groups the introduction of spectral parameters in the twisted yang baxter equation and its solutions are also discussed