Let F be a finite field of q elements and G be a primitive group on a finite set . Then there is a G-action on , namely a map G ! , (g; !) 7! !g = g!; satisfying !gg0 = (gg0)! = g(g0!) for all g; g0 2 G and all ! 2 , and that !1 = 1! = ! for all ! 2 : Let F = ff j f : ! Fg, be the vector space over F with basis . Extending the G-action on linearly, F becomes an FG-module called an FG- permutation module. We are interested in finding all G-invariant FG-submodules, i.e., codes in F . The elements f 2 F are written in the form f = P !2 a! ! where ! is a characteristic function. The natural action of an element g 2 G is given by g P !2 a! ! = P !2 a! g(!): This action of G preserves the natural bilinear form defined by * X a! !; X b! ! + = X a!b!: In this thesis a program is proposed on how to determine codes with given primitive permutation group. The approach is modular representation theoretic and based on a study of maximal submodules of permutation modules F defined by the action of a finite group G on G-sets = G=Gx. This approach provides the advantage of an explicit basis for the code. There appear slightly different concepts of (linear) codes in the literature. Following Knapp and Schmid [83] a code over some finite field F will be a triple (V; ; F), where V = F is a free FG-module of finite rank with basis and a submodule C. By convention we call C a code having ambient space V and ambient basis . F is the alphabet of the code C, the degree n of V its length, and C is an [n; k]-code if C is a free module of dimension k. In this thesis we have surveyed some known methods of constructing codes from primitive permutation representations of finite groups. Generally, our program is more inclusive than these methods as the codes obtained using our approach include the codes obtained using these other methods. The designs obtained by other authors (see for example [40]) are found using our method, and these are in general defined by the support of the codewords of given weight in the codes. Moreover, this method allows for a geometric interpretation of many classes of codewords, and helps establish links with other combinatorial structures, such as designs and graphs. To illustrate the program we determine all 2-modular codes that admit the two known non-isomorphic simple linear groups of order 20160, namely L3(4) and L4(2) = A8. In the process we enumerate and classify all codes preserved by such groups, and provide the lattice of submodules for the corresponding permutation modules. It turns out that there are no self-orthogonal or self-dual codes invariant under these groups, and also that the automorphism groups of their respective codes are in most cases not the prescribed groups. We make use of the Assmus Matson Theorem and the Mac Williams identities in the study of the dual codes. We observe that in all cases the sets of several classes of non-trivial codewords are stabilized by maximal subgroups of the automorphism groups of the codes. The study of the codes invariant under the simple linear group L4(2) leads as a by-product to a unique flag-transitive, point primitive symmetric 2-(64; 28; 12) design preserved by the affi ne group of type 26:S6(2). This has consequently prompted the study of binary codes from the row span of the adjacency matrices of a class of 46 non-isomorphic symmetric 2-(64; 28; 12) designs invariant under the Frobenius group of order 21. Codes obtained from the orbit matrices of these designs have also been studied. The thesis concludes with a discussion of codes that are left invariant by the simple symplectic group S6(2) in all its 2-modular primitive permutation representations. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.

We investigate various methods for testing whether two groups of curves are statistically significantly different, with the motivation to apply the techniques to the analysis of data arising from designed experiments. We propose a set of tests based on pairwise differences between individual curves. Our objective is to compare the power and robustness of a variety of tests, including a collection of permutation tests, a test based on the functional principal components scores, the adaptive Neyman test and the functional F test. We illustrate the application of these tests in the context of a designed 2^4 factorial experiment with a case study using data provided by NASA. We apply the methods for comparing curves to this factorial data by dividing the data into two groups by each effect (A, B, . . . , ABCD) in turn. We carry out a large simulation study investigating the power of the tests in detecting contamination, location, and shift effects on unimodal and monotone curves. We conclude that the permutation test using the mean of the pairwise differences in L1 norm has the best overall power performance and is a robust test statistic applicable in a wide variety of situations. The advantage of using a permutation test is that it is an exact, distribution-free test that performs well overall when applied to functional data. This test may be extended to more than two groups by constructing test statistics based on averages of pairwise differences between curves from the different groups and, as such, is an important building-block for larger experiments and more complex designs.

The minimum degree spanning tree problem is a widely studied NP-hard variation of the minimum spanning tree problem, and a generalization of the Hamiltonian path problem. Most of the work done on the minimum degree spanning tree problem has been on approximation algorithms, and very little work has been done studying graph classes where this problem may be polynomial time solvable. The Hamiltonian path problem has been widely studied on graph classes, and we use classes with polynomial time results for the Hamiltonian path problem as a starting point for graph class results for the minimum degree spanning tree problem. We show the minimum degree spanning tree problem is polynomial time solvable for chain graphs. We then show this problem is polynomial time solvable on bipartite permutation graphs, and that there exist minimum degree spanning trees of these graphs that are caterpillars, and that have other particular structural properties.

O problema de escalonamento num flow shop (ou flow shop scheduling problem, FSSP) é um modelo de sistemas de produção muito comum que é bem estudado na literatura. No entanto, quase toda a literatura foca-se em escalonamentos permutacionais, desconsiderando soluções ótimas e quase ótimas que são escalonamentos não-permutacionais. Além disso, a prática comum padroniza os tempos de processamento de cada operação, mesmo que estes tempos variem dependendo das diferentes capacidades dos operadores das máquinas, cuja diversidade deve ser considerada no processo de escalonamento quando seja significativa, e.g., em centros de emprego para deficientes (CEDs). Nesta tese, propomos métodos para resolver o FSSP não-permutacional, usando o mesmo tempo e esforço que os métodos do estado da arte usam para o FSSP permutacional, e produzindo escalonamentos não-permutacionais com melhor qualidade do que escalonamentos permutacionais e não-permutacionais produzidos por métodos do estado da arte. Também propomos métodos para resolver o problema combinado de designação de trabalhadores heterogêneos e escalonamento de tarefas num flow shop (ou heterogeneous workforce assignment and flow shop scheduling problem, Het-FSSP), produzindo soluções que compensam as diferentes capacidades e deficiências dos trabalhadores com pequenas perdas nos objetivos da produção. Além do mais, a designação de trabalhadores heterogêneos pode ser integrada em outros problemas de escalonamento, como fizemos com o problema combinado de designação de trabalhadores heterogêneos e escalonamento de tarefas num job shop (ou heterogeneous workforce assignment and job shop scheduling problem, Het-JSSP). / The flow shop scheduling problem (or FSSP) is a very common model of production systems that is well studied in the literature. However, almost all the literature focuses on the permutation FSSP, disregarding optimal and near optimal solutions that are non-permutation schedules. Besides, common practice standardizes the processing times of each operation, even when those times may vary depending on different capabilities of the machine operators, whose diversity must be considered in the scheduling process when it is significant, e.g., in Sheltered Work centers for Disabled (SWDs). In this thesis, we propose methods to solve the non-permutation FSSP, using the same time and effort as state-of-the-art methods for the permutation FSSP, and producing non-permutation schedules with better quality than permutation and non-permutation schedules produced by state-of-the-art methods. We also propose methods to solve the combined heterogeneous workforce assignment and flow shop scheduling problem (or Het-FSSP), producing solutions that compensate the different capabilities and disabilities of the workers with minor or null losses in the productivity objectives. Moreover, the heterogeneous workforce assignment may be integrated into other shop scheduling models, as we did with the heterogeneous workforce assignment and job shop scheduling problem (or Het-JSSP) with similar results.

The current thesis is focused on synchronizing permutation groups and on graph endo- morphisms. Applying the implicit classification of rank 3 groups, we provide a bound on synchronizing ranks of rank 3 groups, at first. Then, we determine the singular graph endomorphisms of the Hamming graph and related graphs, count Latin hypercuboids of class r, establish their relation to mixed MDS codes, investigate G-decompositions of (non)-synchronizing semigroups, and analyse the kernel graph construction used in the theorem of Cameron and Kazanidis which identifies non-synchronizing transformations with graph endomorphisms [20]. The contribution lies in the following points: 1. A bound on synchronizing ranks of groups of permutation rank 3 is given, and a complete list of small non-synchronizing groups of permutation rank 3 is provided (see Chapter 3). 2. The singular endomorphisms of the Hamming graph and some related graphs are characterised (see Chapter 5). 3. A theorem on the extension of partial Latin hypercuboids is given, Latin hyper- cuboids for small values are counted, and their correspondence to mixed MDS codes is unveiled (see Chapter 6). 4. The research on normalizing groups from [3] is extended to semigroups of the form < G, T >, and decomposition properties of non-synchronizing semigroups are described which are then applied to semigroups induced by combinatorial tiling problems (see Chapter 7). 5. At last, it is shown that all rank 3 graphs admitting singular endomorphisms are hulls and it is conjectured that a hull on n vertices has minimal generating set of at most n generators (see Chapter 8).

O problema de escalonamento num flow shop (ou flow shop scheduling problem, FSSP) é um modelo de sistemas de produção muito comum que é bem estudado na literatura. No entanto, quase toda a literatura foca-se em escalonamentos permutacionais, desconsiderando soluções ótimas e quase ótimas que são escalonamentos não-permutacionais. Além disso, a prática comum padroniza os tempos de processamento de cada operação, mesmo que estes tempos variem dependendo das diferentes capacidades dos operadores das máquinas, cuja diversidade deve ser considerada no processo de escalonamento quando seja significativa, e.g., em centros de emprego para deficientes (CEDs). Nesta tese, propomos métodos para resolver o FSSP não-permutacional, usando o mesmo tempo e esforço que os métodos do estado da arte usam para o FSSP permutacional, e produzindo escalonamentos não-permutacionais com melhor qualidade do que escalonamentos permutacionais e não-permutacionais produzidos por métodos do estado da arte. Também propomos métodos para resolver o problema combinado de designação de trabalhadores heterogêneos e escalonamento de tarefas num flow shop (ou heterogeneous workforce assignment and flow shop scheduling problem, Het-FSSP), produzindo soluções que compensam as diferentes capacidades e deficiências dos trabalhadores com pequenas perdas nos objetivos da produção. Além do mais, a designação de trabalhadores heterogêneos pode ser integrada em outros problemas de escalonamento, como fizemos com o problema combinado de designação de trabalhadores heterogêneos e escalonamento de tarefas num job shop (ou heterogeneous workforce assignment and job shop scheduling problem, Het-JSSP). / The flow shop scheduling problem (or FSSP) is a very common model of production systems that is well studied in the literature. However, almost all the literature focuses on the permutation FSSP, disregarding optimal and near optimal solutions that are non-permutation schedules. Besides, common practice standardizes the processing times of each operation, even when those times may vary depending on different capabilities of the machine operators, whose diversity must be considered in the scheduling process when it is significant, e.g., in Sheltered Work centers for Disabled (SWDs). In this thesis, we propose methods to solve the non-permutation FSSP, using the same time and effort as state-of-the-art methods for the permutation FSSP, and producing non-permutation schedules with better quality than permutation and non-permutation schedules produced by state-of-the-art methods. We also propose methods to solve the combined heterogeneous workforce assignment and flow shop scheduling problem (or Het-FSSP), producing solutions that compensate the different capabilities and disabilities of the workers with minor or null losses in the productivity objectives. Moreover, the heterogeneous workforce assignment may be integrated into other shop scheduling models, as we did with the heterogeneous workforce assignment and job shop scheduling problem (or Het-JSSP) with similar results.

O problema de escalonamento num flow shop (ou flow shop scheduling problem, FSSP) é um modelo de sistemas de produção muito comum que é bem estudado na literatura. No entanto, quase toda a literatura foca-se em escalonamentos permutacionais, desconsiderando soluções ótimas e quase ótimas que são escalonamentos não-permutacionais. Além disso, a prática comum padroniza os tempos de processamento de cada operação, mesmo que estes tempos variem dependendo das diferentes capacidades dos operadores das máquinas, cuja diversidade deve ser considerada no processo de escalonamento quando seja significativa, e.g., em centros de emprego para deficientes (CEDs). Nesta tese, propomos métodos para resolver o FSSP não-permutacional, usando o mesmo tempo e esforço que os métodos do estado da arte usam para o FSSP permutacional, e produzindo escalonamentos não-permutacionais com melhor qualidade do que escalonamentos permutacionais e não-permutacionais produzidos por métodos do estado da arte. Também propomos métodos para resolver o problema combinado de designação de trabalhadores heterogêneos e escalonamento de tarefas num flow shop (ou heterogeneous workforce assignment and flow shop scheduling problem, Het-FSSP), produzindo soluções que compensam as diferentes capacidades e deficiências dos trabalhadores com pequenas perdas nos objetivos da produção. Além do mais, a designação de trabalhadores heterogêneos pode ser integrada em outros problemas de escalonamento, como fizemos com o problema combinado de designação de trabalhadores heterogêneos e escalonamento de tarefas num job shop (ou heterogeneous workforce assignment and job shop scheduling problem, Het-JSSP). / The flow shop scheduling problem (or FSSP) is a very common model of production systems that is well studied in the literature. However, almost all the literature focuses on the permutation FSSP, disregarding optimal and near optimal solutions that are non-permutation schedules. Besides, common practice standardizes the processing times of each operation, even when those times may vary depending on different capabilities of the machine operators, whose diversity must be considered in the scheduling process when it is significant, e.g., in Sheltered Work centers for Disabled (SWDs). In this thesis, we propose methods to solve the non-permutation FSSP, using the same time and effort as state-of-the-art methods for the permutation FSSP, and producing non-permutation schedules with better quality than permutation and non-permutation schedules produced by state-of-the-art methods. We also propose methods to solve the combined heterogeneous workforce assignment and flow shop scheduling problem (or Het-FSSP), producing solutions that compensate the different capabilities and disabilities of the workers with minor or null losses in the productivity objectives. Moreover, the heterogeneous workforce assignment may be integrated into other shop scheduling models, as we did with the heterogeneous workforce assignment and job shop scheduling problem (or Het-JSSP) with similar results.

Let p be a prime, p a power of p and 𝔽q the finite field with q elements. Any function φ: 𝔽q → 𝔽q can be unqiuely represented by a polynomial, 𝔽φ of degree < q. If the map x ↦ Fφ(x) induces a permutation on the underlying field we say Fφ is a permutation polynomial. Permutation polynomials have applications in many diverse fields of mathematics. In this dissertation we are generally concerned with the following question: Given a polynomial f, when does the map x ↦ F(x) induce a permutation on 𝔽q. In the second chapter we are concerned the permutation behavior of the polynomial gn,q, a q-ary version of the reversed Dickson polynomial, when the integer n is of the form n = qa - qb - 1. This leads to the third chapter where we consider binomials and trinomials taking special forms. In this case we are able to give explicit conditions that guarantee the given binomial or trinomial is a permutation polynomial. In the fourth chapter we are concerned with permutation polynomials of 𝔽q, where q is even, that can be represented as the sum of a power function and a linearized polynomial. These types of permutation polynomials have applications in cryptography. Lastly, chapter five is concerned with a conjecture on monomial graphs that can be formulated in terms of polynomials over finite fields.

In this paper, new variations of some well-known permutation statistics are introduced and studied. Firstly, a double-descent of a permutation π is defined as a position i where πi ≥ 2πi+1. By proofs by induction and direct proofs, recursive and explicit expressions for the number of n-permutations with k double-descents are presented. Also, an expression for the total number of double-descents in all n-permutations is presented. Secondly, a double-inversion of a permutation π is defined as a pair (πi,πj) where i&lt;j but πi ≥ 2πj. The total number of double-inversions in all n-permutations is presented.

A superpattern is a string of characters of length n over [k]={1, 2, …, k} that contains as a subsequence, and in a sense that depends on the context, all the smaller strings of length k in a certain class. We prove structural and probabilistic results on superpatterns for preferential arrangements, including (i) a theorem that demonstrates that a string is a superpattern for all preferential arrangements if and only if it is a superpattern for all permutations; and (ii) a result that is reminiscent of a still unresolved conjecture of Alon on the smallest permutation on [n] that contains all k-permutations with high probability.