Combinatorial Rules Research Articles

Molecular solvent design for liquid–liquid extraction using the UNIQUAC model

In this work molecular solvent design for liquid–liquid extraction using a chemical graphics system is presented. Solvent structures are generated according to specified combinatorial rules and the screening of the solvent is performed using the UNIQUAC model where the required molecular interaction parameters are directly obtained from a molecular graphics system. The method is tested with two different industrial examples: toluene/ n-heptane and phenol/water system.

Comments on "Combinatorial rule explosion eliminated by a fuzzy rule configuration" [with reply

In the original paper (IEEE Trans. Fuzzy Syst., vol.6, p.1-11, 1998), Combs and Andrews proved the following logical equivalence (stated here for two antecedents p and q and one consequent r, but easily generalize to an arbitrary number of antecedents and consequents): [(p/spl and/q)/spl rArr/r]/spl hArr/[(p/spl rArr/r)V(q/spl rArr/r)]. This is a very significant result because it suggests that we can replace multi-antecedent rules with an interconnection of single antecedent rules, which eliminates the rule explosion that is associated with multi-antecedent rules. Combs and Andrews refer to the left-hand side of this equivalence as an intersection rule configuration (IRC) and to its right-hand side as a union rule configuration (URC). Their result gives rise to two distinctly different paths for the design of fuzzy logic systems; IRC, which leads to rule explosion, and URC, which does not. The authors discuss four points about the IRC/spl hArr/URC relation. The original authors reply, acknowledging some of the points and stating that they would present their results differently if starting now.

Scheduling grouped jobs on single machine with genetic algorithm

Production scheduling of grouped jobs has been an active research area since GT (Group Technology) was widely applied in practical manufacturing systems. To minimize the total flowtime of grouped jobs on a single machine, we combine jobs into fundamental runs based upon the necessary condition of the optimal solution. It is proved that the optimal solution is a combination of fundamental runs. A genetic algorithm is designed based on studies on the combinatorial rules of fundamental runs. The numerical results show that the computational performance of the algorithm depends on the number of ‘fundamental’ runs, not on the number of jobs. In general, the number of fundamental runs is far less than the number of jobs. Therefore, the algorithm has potential for practical application in large scale production systems.

Diagram Rules for the Generation of Schubert Polynomials

We prove an elegant combinatorial rule for the generation of Schubert polynomials based on box diagrams, which was conjectured by A. Kohnert. The main tools for the proof are (1) a recursive structure of Schubert polynomials and (2) a partial order on the set of box diagrams. As a byproduct we obtain (combinatorial) proofs for two other rules for the generation of Schubert polynomials based on box diagrams: (1) the more complicated rule of N. Bergeron, and (2) the rule of P. Magyar, which we show to be a simplified Bergeron rule. The well-known fact that the Schubert polynomials associated to Grassmannian permutations are in fact Schur polynomials is derived from Kohnert's rule.

A Littlewood-Richardson rule for factorial Schur functions

We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.

Solvent Design for Liquid Extraction Using Calculated Molecular Interaction Parameters

In this work, a molecular solvent design procedure for separation processes such as liquidliquid extraction, absorption, etc. is presented. Solvent molecular structures built from chosen sets of groups are generated in the three dimensional space by the use of a molecular graphics system (MGS) according to user specified combinatorial rules. The screening of the generated solvent candidates is performed according to distribution coefficients, selectivities, capacities, solvent losses etc. and which are the primary and most important criteria for, particularly, liquid-liquid extraction. The design procedure is based upon the UNIFAC model estimates of the activity coefficients at infinite dilution of the different compounds present in the extract and raffinate phases, where calculated molecular, rather than the existing liquid-liquid group interaction parameters, are used. The method is tested with two different industrial examples namely toluene/n-heptane and phenol/water systems.

The impact of priority rule combinations on lateness and tardiness

Several recent studies have explored the concurrent deployment of different priority rules at different processing stages of a manufacturing system. This study investigates the same issue by combining three popular simple priority rules with a combinatorial rule. In a three-stage flow shop, these rules are combined into 64 combination schemes and their performance compared under two shop load levels with two due date setting methods. The performance criteria considered are: mean lateness; mean tardiness; maximum tardiness; and per cent of tardy jobs. The results indicate that the rule combinations are a better strategy than their pure forms when various performance measures are jointly evaluated. Particularly, selected combinations of the Modified Shortest Processing Time, Shortest Processing Time, and the Earliest Due Date rules appear to be very effective. While the extent of shop load level shows little impact on the relative performance of the schemes, the endogenous method of due date setting consistently yields better results than the exogenous method.

Demazure characters and WZW fusion rules

The Demazure character formula is applied to the Verlinde formula for the fusion rules of Wess–Zumino–Witten (WZW) models. We follow Littelmann’s derivation of a generalized Littlewood–Richardson rule that computes tensor products for simple Lie algebras. A combinatorial rule for WZW fusions does not result, however. Only a modified version of the Littlewood–Richardson rule is obtained that computes an (old) upper bound on the fusion coefficients of Ar WZW models. We argue that this is because the characters of simple Lie algebras appear in this treatment, instead of the corresponding affine characters. The Bruhat order on the affine Weyl group must be implicated in any combinatorial rule for WZW fusions; the Bruhat order on subgroups of this group (such as the finite Weyl group) does not suffice.

On the Multiplication of Schubert Polynomials

Finding a combinatorial rule for the multiplication of Schubert polynomials is a long standing problem. In this paper we give a combinatorial proof of the extended Pieri rule as conjectured by N. Bergeron and S. Billey, which says how to multiply a Schubert polynomial by a complete or elementary symmetric polynomial, and describe some observations in the direction of a general rule.

Combinatorial rule explosion eliminated by a fuzzy rule configuration

Conventional fuzzy inference methodology relates the relevant subsets of each input universal set to the subsets of the other system inputs through an intersection-rule configuration. This strategy yields an exponential growth in the number of rules as inputs are added to the system, quickly reducing performance to unacceptable levels. A novel rule configuration and matrix design are presented in this paper that do not rely on rule multiplication to insure that antecedent elements are effectively related to their consequent counterparts. This alternative formulation models the entire system problem space with a simplified structure that increases linearly as the inference engine grows, providing significant computational savings to a broad range of commercial and scientific applications.

Co-response Coefficients, Monovalent Units, and Combinatorial Rules: Unification of Concepts in Metabolic Control Analysis

The JacobianHof a linear metabolic pathway without feedback loops is tridiagonal. Its inverse,H−1, which is needed for calculating control coefficients or elasticities, can be decomposed into two regions of mutually dependent rows and columns. For each of these regions ofH−1, all sub-determinants of order two are zero. The existence of the two regions is shown to cause certain invariance properties of the ratios of control coefficients that can be expressed as co-response coefficients. Also the concept of monovalent functional units seems to be related to the existence of the regions. Moreover, the combinatorial rules for selecting the right modulations originate from the fact that certain sub-determinants ofH−1are zero when located within the regions. The regions of zero sub-determinants of order two ofH−1are thus a central element for the analysis of the linear pathway. The unifying potential ofH−1is not restricted to the linear chain but is also expected to be valid for complex metabolic systems.

Extending Double Modulation: Combinatorial Rules for Identifying the Modulations Necessary for Determining Elasticities in Metabolic Pathways

The double modulation method for determining the elasticities of pathway enzymes, originally devised by Kacser & Burns ( Biochem. Soc. Trans. 7,1149–1160, 1979), is extended to pathways of complex topological structure, including branching and feedback loops. An explicit system of linear equations for the unknown elasticities is derived. The constraints imposed on this linear system imply that modulations of more than one enzyme are not necessarily independent. Simple combinatorial rules are described for identifying without using any algebra the set of independent modulations that allow the determination of the elasticities of any enzyme. By repeated application, the minimum numbers of modulations required to determine the elasticities of all enzymes of a given pathway can be determined. The procedure is illustrated with numerous examples.

Behavioural knowledge integration in the design studio: an experimental evaluation of three strategies

This study addresses the difficulties in design education regarding scientific knowledge generation and its application in the design studio by investigating the effects of particular design approaches on the integration of sociobehavioural scientific information by design students. The investigation involved the systematic assignment of the students to one of three treatments representing alternative design approaches and assessment of the effects of these treatments on the quality of their design products in terms of knowledge integration. The three treatments approximate the major methodological approaches that are debated in the literature. They differ in their assumptions on the nature of the design process, the form of knowledge, and the type of information representation. The results support the main hypothesis that a treatment corresponding to the integrative/ generative strategy, in which ‘tyle’ (a form image, combinatorial rules and a theoretical framework) serves as a primary generator, is superior to a listing of empirical information generated by an external source. Given the discussion in the literature on the design process and cognition and the results of this study, it is possible to state that the applicability gap is expected to diminish to the extent that the form of knowledge embodied in ‘style’ is explicitly linked to the development of scientific knowledge.

Characters of Algebras Containing a Jones Basic Construction: The Temperley-Lieb, Okada, Brauer, and Birman-Wenzl Algebras

We begin by determining, in a general form, the characters of irreducible representations of a Jones basic construction and use this result to compute the characters of the Temperley-Lieb algebras and the Okada algebras. In the case of the Birman-Wenzl algebra some of the characters are determined by the general theorem and the others are computed by using the duality between the Birman-Wenzl algebras and the Drinfeld-Jimbo quantum groups of types B and C. The computations involve certain characters of the quantum group; these are polynomials invariant under the Weyl group of type B. We are able to decompose these Weyl group symmetric functions and obtain a combinatorial rule for computing the irreducible characters of the Birman-Wenzl algebras. The combinatorial rules for computing the irreducible characters of the Iwahori-Hecke algebras and the combinatorial rule for computing the irreducible characters of the Brauer algebras are both special cases of our rule for the Birman-Wenzl algebras.

A combinatorial rule for the schur function expansion of the PlethysmS(1a,b)[Pk

We present a simple combinatorial rule to expand the plethysm Pn [S(1 a ,b)](x) of a power summetric function Pn (x) and a Schur function of hook shapeS(1 a ,b)(x), as a sum of Schur functions. The key ingredient of our proof is a correspondence between the circle diagrams, introduced by Chen, Garsia and Remmel [6] in their SXP algorithm to compute the Schur function expansion of Pn [S λ], and certain special rim hook and transposed special rim hook tabloids which is of interest in its own right. As an application of our rule, we drive explicit formulas for the coefficient of any Schur function of hook shape in the Schur function expansion of a plethysm of any two Schur functions of hook shape.

A q-rational Murnaghan-Nakayama rule

The Murnaghan-Nakayama rule is a combinatorial rule for computing symmetric group characters. It has recently been extended to compute Iwahori-Hecke algebra characters and Brauer algebra characters. It is proved using the fact that the symmetric group L f and the general linear group GL( r, C ) centralize each other on the tensor space ⊗ f V of f copies of the natural representation V of GL( r, C ). This tensor space contains the polynomial representations of GL( r, C ). The mixed tensor space T m, n = (⊗ m V) ⊗ (⊗ n V∗, where V∗ is the dual to V, contains the rational representations of GL( r, C ). When r ⩾ m + n, its centralizer is the complex algebra B m, n r which is a subalgebra of the Brauer algebra B m + n r . If we let V q = V ⊗ C ( q), then T q m, n = (⊗ m V q ) ⊗ (⊗ n V q ∗ ) contains the rational representations of the quantum group U q ( gl( r, C )), and when r ⩾ m + n its centralizer is the two-parameter Iwahori-Hecke algebra H m, n r ( q). We derive a combinatorial rule for computing B m, n r -characters and H m, n r ( q)-characters, and use it to compute character tables. The Murnaghan-Nakayama rules for L m and the Iwahori-Hecke algebra H m ( q) are obtained as the special case when n = 0.

Characters of Brauer’s centralizer algebras

Brauer's centralizer algebras are finite dimensional algebras with a distinguished basis. Each Brauer centralizer algebra contains the group algebra of a symmetric group as a subalgebra and the distinguished basis of the Brauer algebra contains the permutations as a subset. In view of this containment it is desirable to generalize as many known facts concerning the group algebra of the symmetric group to the Brauer algebras as possible. This paper studies the irreducible characters of the Brauer algebras in view of the distinguished basis. In particular we define an analogue of conjugacy classes, and derive Frobenius formulas for the characters of the Brauer algebras. Using the Frobenius formulas we derive formulas for the irreducible character of the Brauer algebras in terms of the irreducible characters of the symmetric groups and give a combinatorial rule for computing these irreducible characters.

A generalization of solomon’s algebra for hyperoctahedral groups and other wreath products

In this paper we give a combinatorial rule to compute the composition of two convolution products of endomorphisms of a free associative algebra and deduce the construction of a subalgebra of QB n ...

A combinatorial rule to determine the probability of n consecutive successes of N tries

A procedure for the determination of the probability or n consecutive successes of N tries is presented The procedure is based on the formation of a coefficient array. The coefficients are determined using a combinatorial rule.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A new adaptively weighted combinatorial dispatching rule for complex scheduling problems

In this paper, a new adaptively weighted combinatorial dispatching (AWCD) rule is developed which is appropriate for complex scheduling problems with multiple and conflicting criteria. This rule identifies the constraints that are most tight, and adaptively estimates the tightness of each constraint. Previous combinatorial dispatching rules appear as special cases of the AWCD rule. An illustrative small-scale scheduling example is provided, which shows the applicability of the AWCD rule to practical problems. Statistical results obtained using 200 random scheduling problems showed the superiority of the AWCD rule over six other composite or combinatorial rules.