Multi-component reactions between 2-aminopyrimidine, aldehydes and isonitriles: the use of a nonpolar solvent suppresses formation of multiple products

A reliable and convenient protocol for MCRs between 2-aminopyrimidine, aldehydes and isonitriles is described. Toluene was chosen as the reaction solvent based on mechanistic reasoning. The product mixtures were found to contain a single imidazo[1,2-a]pyrimidine product in each case, which were isolated and purified by precipitation and crystallization. Chromatographic recovery from filtrates raised the total yields to 49-66%. The procedure provides a practical alternative to hitherto reported methods in which formation of multiple products was observed.     

A mathematical programming model and solution for scheduling production orders in Shanghai Baoshan Iron and Steel Complex

In this paper, we investigate the production order scheduling problem derived from the production of steel sheets in Shanghai Baoshan Iron and Steel Complex (Baosteel). A deterministic mixed integer programming (MIP) model for scheduling production orders on some critical and bottleneck operations in Baosteel is presented in which practical technological constraints have been considered. The objective is to determine the starting and ending times of production orders on corresponding operations under capacity constraints for minimizing the sum of weighted completion times of all orders. Due to large numbers of variables and constraints in the model, a decomposition solution methodology based on a synergistic combination of Lagrangian relaxation, linear programming and heuristics is developed. Unlike the commonly used method of relaxing capacity constraints, this methodology alternatively relaxes constraints coupling integer variables with continuous variables which are introduced to the objective function by Lagrangian multipliers. The Lagrangian relaxed problem can be decomposed into two sub-problems by separating continuous variables from integer ones. The sub-problem that relates to continuous variables is a linear programming problem which can be solved using standard software package OSL, while the other sub-problem is an integer programming problem which can be solved optimally by further decomposition. The subgradient optimization method is used to update Lagrangian multipliers. A production order scheduling simulation system for Baosteel is developed by embedding the above Lagrangian heuristics. Computational results for problems with up to 100 orders show that the proposed Lagrangian relaxation method is stable and can find good solutions within a reasonable time.     

The one dimensional Compartmentalised Knapsack Problem: A case study

The Compartmentalised Knapsack Problem (CKP) is similar to the ordinary Knapsack Problem except that items to be packed belong to separate classes, and items can only be packed, in knapsack compartments, amongst items in their own class. This paper addresses a case study in the cutting of steel rolls in which the CKP arises. The rolls are cut in two-phases: the first phase produces sub-rolls (compartments) which are, after reducing the thickness, cut in a second phase to produce ribbons (a class consists of ordered items with the same thickness). Finally, two methods of solving CKP are presented, and these are used to generate columns in the classical linear optimisation model of Gilmore and Gomory. Results of computational experiments are presented.     

Applications of residues to combinatorial identities

A concrete aspect of Grothendieck Duality is used to give local cohomology proofs of combinatorial identities including MacMahon's master theorem, Grosswald identity, identity of Shoo, Tepper identity, and others.

     

Xanthines as a scaffold for molecular diversity

Summary Xanthines represent a new, versatile scaffold for combinatorial chemistry. A five-step solid-phase synthesis of xanthine derivatives is described which includes alkylations, a nucleophilic displacement reaction at a heterocycle and a ring closure reaction by condensation of a nitroso function with an activated methylene group. The selected reaction sequence allows the production of a highly diverse small-molecule combinatorial compound library.     

Solving the maximum edge weight clique problem via unconstrained quadratic programming

The unconstrained quadratic binary program (UQP) is proving to be a successful modeling and solution framework for a variety of combinatorial optimization problems. Experience reported in the literature with several problem classes has demonstrated that this approach works surprisingly well in terms of solution quality and computational times, often rivaling and sometimes surpassing more traditional methods. In this paper we report on the application of UQP to the maximum edge-weighted clique problem. Computational experience is reported illustrating the attractiveness of the approach.     

Combinatorial data analysis: Confirmatory comparisons between sets of matrices

The task of assessing the similarity of pattern between the entries of two square matrices has been discussed extensively over the last decade, as a unifying strategy for approaching a variety of seemingly disparate statistical problems. As typically defined, the comparison depends on a measure of matrix correspondence, usually a normalized cross-product measure of some form, that is evaluated for relative size by the use of a reference distribution constructed through an equally likely permutation hypothesis defined at the level of the objects corresponding to the rows and columns of the two matrices. The extreme generality provided by this very simple framework subsumes a variety of different statistical problems, ranging from the study of spatial autocorrelation for variables observed over a set of geographic locations, to the topics of analysis of variance, the measurement of rank correlation, and confirmation techniques concerned with various conjectures of combinatorial structure that might be posited for an empirically determined measure of relationship between pairs of a given set of objects. The comparison strategies extant always assume that both matrices are fixed, and in those cases where one of the matrices codifies a given theoretical structure to be evaluated according to a second, this assumption can lead to substantial arbitrariness in how matrix similarity might be indexed, and thus, in how the comparison is implemented. As developed in this paper, exactly the same principles appropriate for use in the fixed comparison context can be extended to include matrices constructed through optimally weighted linear combinations of other sets of matrices. This generalization provides one mechanism for developing comparison strategies that allow assessment against very broad classes of matrices, which in turn serve to represent very general conjectures of possible combinatorial structure. This paper reviews some of these extensions in detail, with a particular emphasis on categorical and ordered categorical variables and whether they may reflect an empirically generated measure of object relationship.     

A reciprocity theorem for certain q-series found in Ramanujan’s lost notebook

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s ₁ψ₁ summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D15 B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency. A. J. Yee: Research partially supported by a grant from The Number Theory Foundation.     

Discrete optimization in public rail transport

Many problems arising in traffic planning can be modelled and solved using discrete optimization. We will focus on recent developments which were applied to large scale real world instances. Most railroad companies apply a hierarchically structured planning process. Starting with the definition of the underlying network used for transport one has to decide which infrastructural improvements are necessary. Usually, the rail system is periodically scheduled. A fundamental base of the schedule are the lines connecting several stations with a fixed frequency. Possible objectives for the construction of the line plan may be the minimization of the total cost or the maximization of the passengers’s comfort satisfying certain regulations. After the lines of the system are fixed, the train schedule can be determined. A criterion for the quality of a schedule is the total transit time of the passengers including the waiting time which should be minimized satisfying some operational constraints. For each trip of the schedule a train consisting of a locomotive and some carriages is needed for service. The assignment of rolling stock to schedule trips has to satisfy operational requirements. A comprehensible objective is to minimize the total cost. After all strategic and tactical planning the schedule has to be realized. Several external influences, for example delayed trains, force the dispatcher to recompute parts of the schedule on-line. A Web page with examples quoted in this survey can be found at http://www.math.tu-bs.de/mo/ismp.html.     

The application of automated reasoning to formal models of combinatorial optimization

Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynamic programming, branch and bound, and greedy. In 1989 Helman [9] presented a common formalism that captures dynamic programming and branch and bound type algorithms. The formalism was later extended to include greedy algorithms. In this paper, we describe the application of automated reasoning techniques to the domain of our model, in particular considering some representational issues and demonstrating that proofs about the model can be obtained by an automated reasoning program. The long-term objective of this research is to develop a methodology for using automated reasoning to establish new results within the theory, including the derivation of new lower bounds and the discovery (and verification) of new combinatorial search strategies.