On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem

Pólya’s fundamental enumeration theorem and some results from Williamson’s generalized setup of it are proved in terms of Schur- Macdonald’s theory (S-MT) of “invariant matrices”. Given a permutation group W ≤ Sd and a one-dimensional character χ of W , the polynomial functor Fχ corresponding via S-MT to the induced monomial representation Uχ = ind|Sdv/W (χ) of Sd , is studied. It turns out that the characteristic ch(Fχ ) is the weighted inventory of some set J(χ) of W -orbits in the integer-valued hypercube [0, ∞)d . The elements of J(χ) can be distinguished among all W -orbits by a maximum property. The identity ch(Fχ ) = ch(Uχ ) of both characteristics is a consequence of S-MT, and is equivalent to a result of Williamson. Pólya’s theorem can be obtained from the above identity by the specialization χ = 1W , where 1W is the unit character of W.

Isomerism as Manifestation of Intrinsic Symmetry of Molecules: Lunn–Senior’s Theory

This article presents the principal results of the doctoral thesis “Isomerism as internal symmetry of molecules” by Valentin Vankov Iliev (Institute of Mathematics and Informatics), successfully defended before the Specialised Academic Council for Informatics and Mathematical Modelling on 15 December, 2008.

Learning Technology in Scheduling Based on the Mixed Graphs

We propose the adaptive algorithm for solving a set of similar scheduling problems using learning technology. It is devised to combine the merits of an exact algorithm based on the mixed graph model and heuristics oriented on the real-world scheduling problems. The former may ensure high quality of the solution by means of an implicit exhausting enumeration of the feasible schedules. The latter may be developed for certain type of problems using their peculiarities. The main idea of the learning technology is to produce effective (in performance measure) and efficient (in computational time) heuristics by adapting local decisions for the scheduling problems under consideration. Adaptation is realized at the stage of learning while solving a set of sample scheduling problems using a branch-and-bound algorithm and structuring knowledge using pattern recognition apparatus.

On Solving the Maximum Betweenness Problem Using Genetic Algorithms

In this paper a genetic algorithm (GA) is applied on Maximum Betweennes Problem (MBP). The maximum of the objective function is obtained by finding a permutation which satisfies a maximal number of betweenness constraints. Every permutation considered is genetically coded with an integer representation. Standard operators are used in the GA. Instances in the experimental results are randomly generated. For smaller dimensions, optimal solutions of MBP are obtained by total enumeration. For those instances, the GA reached all optimal solutions except one. The GA also obtained results for larger instances of up to 50 elements and 1000 triples. The running time of execution and finding optimal results is quite short.

Variable Neighborhood Search for the File Transfer Scheduling Problem

ACM Computing Classification System (1998): I.2.8, G.1.6.

Fraction Space Revisited

Бойко Бл. Банчев - Знае се, че рационалните числа образуват интересни и богати на изчислителни възможности структури като редици на Фарей (Феъри) и безкрайни дървета. Малко внимание се обръща на по-общо, систематично излагане на основните свойства на дробите като множество. Понятия биват въвеждани без обосноваване, някои доказателства са ненужно изкуствени, а почти винаги и едните, и другите като че биват отнесени към една или друга особена структура, вместо към множеството на дробите изобщо. Изненадващо е, че някои същностни твърдения изглежда дори не са формулирани в литературата по теория на числата. Тази статия има за цел да подобри състоянието на нещата в това отношение, като предлага общо, подходящо подредено изложение на понятия и свързани с тях твърдения. Като допълнение са представени бележки върху пораждането на множеството от всички дроби – откритие значително по-старо, отколкото е прието да се смята.

Constructing 7-Clusters

ACM Computing Classification System (1998): G.2, G.4.