215 research outputs found
Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups
An partial Latin rectangle is an matrix containing elements of such that each row and each column contain at most one copy of any symbol in . An entry is a triple with . Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively. We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of -entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) , i.e., partial Latin squares, (b) and , and (c) and
Generalized Semimagic Squares for Digital Halftoning
Completing Aronov et al.'s study on zero-discrepancy matrices for digital
halftoning, we determine all (m, n, k, l) for which it is possible to put mn
consecutive integers on an m-by-n board (with wrap-around) so that each k-by-l
region holds the same sum. For one of the cases where this is impossible, we
give a heuristic method to find a matrix with small discrepancy.Comment: 6 pages, 6 figure
Multiplicative and Exponential Variations of Orthomorphisms of Cyclic Groups
An orthomorphism is a permutation of for which
is also a permutation on . Eberhard,
Manners, Mrazovi\'c, showed that the number of such orthomorphisms is
for odd and zero otherwise.
In this paper we prove two analogs of these results where is
replaced by (a "multiplicative orthomorphism") or with
(an "exponential orthomorphism"). Namely, we show that no
multiplicative orthomorphisms exist for but that exponential
orthomorphisms exist whenever is twice a prime such that is
squarefree. In the latter case we then estimate the number of exponential
orthomorphisms.Comment: 11 pages, 1 figur
Selected introductory concepts from combinatorial mathematics, 1967
This paper is concerned with the development of that part of combinatorial mathematics that deals with existence-type problems. This development is accomplished through the framework of modern algebra. Beginning with such elementary tonics as sets, permutations, and combinations the paper goes on to the principle of Inclusion and exclusion, recurrence relations, the elegant Theorem of Ramsey, and an introduction to systems of distinct representatives. In addition to the treatment of combinatorial mathematics as a mathematical system in itself, a few of the multitudinous applications of this theory are presented. These include applications and relationships to the theory of numbers, matrices, group and field theory, and conbiaatorlal-type problems which occur in every day life
Comparing 1st and 4th Grade Curriculum of the United States and Uruguay
Comparing 1st and 4th Grade Curriculum of the United States and Uruguay
A historical perspective of the theory of isotopisms
In the middle of the twentieth century, Albert and Bruck introduced the theory of isotopisms of non-associative algebras and quasigroups as a generalization of the classical theory of isomorphisms in order to study and classify such structures according to more general symmetries. Since then, a wide range of applications have arisen in the literature concerning the classification and enumeration of different algebraic and combinatorial structures according to their isotopism classes. In spite of that, there does not exist any contribution dealing with the origin and development of such a theory. This paper is a first approach in this regard.Junta de AndalucÃ
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