Intercalates and the maximum number of intercalates are presented. We introduced partially intercalate complete Latin squares and results on the existence of some infinite families as well as Latin squares of smaller size is given. The second part of our work summarizes the main results on orthogonal Latin squares. Special type of Latin squares, gerechte designs, are introduced and proof for the existence of such orthogonal Latin squares of prime orders is also presented.

<p> The insertion of lithium into the rutile structures RuO 2 and IrO2 was found, by X-ray powder diffraction, to result in the formation of the orthorhombic phases Li0.9RuO2 and Li0.9IrO2, respectively. The orthorhombic cells of the intercalation products are related to the tetragonal transition metal dioxide cells by a large expansion (~ 0.5 Å) in the [100] and [010] directions and a smaller contraction (~ 0.3 Å for Li0.9RuO2 and 0.05 Å for Li0.9IrO2) along the [100] axis. Both LixRuO2 and LixIrO2 are two phased systems over the range of lithium compositions studied (0.2 <~ x <~ 0.9).</p> <p> The structure of Li0.9RuO2 was determined by neutron powder diffraction. The structure adopted by Li0.9RuO2 is a distorted NiAs-type structure quite similar to that of a related compound LiMoO2. Unresolved difficulties were encountered in attempting to prepare samples of LixOsO2 and LixIrO2 suitable for their structure determination.</p> / Thesis / Master of Science (MSc)

This dissertation includes work done on graphene and related materials, examining their electronic and topological properties using first-principles methods. Ab-initio computational methods, like density functional theory (DFT), have become increasingly popular in condensed matter and material science. Motivated by the search for novel materials that would help us devise fast, low-power, post-CMOS transistors, we explore the properties of some of these promising materials. We begin by studying graphene and its interaction with dielectric oxides. Graphene has recently inspired a flurry of research activity due to its interesting electronic and mechanical properties. For the device community, graphene's high charge carrier mobility and continuous gap tunability can have immense use in novel transistors. In Chapter 3 we examine the properties of graphene placed on two oxides, namely quartz and alumina. We find that oxygen-terminated quartz is a useful oxide for the purpose of graphene based FETs. Inspired by a recent surge of interest in topological insulators, we then explore the topological properties of two-dimensional materials. We conduct a theoretical study to examine the relationship between crystal space group symmetry and the electric polarization of a two-dimensional crystal. We show that the presence of symmetry restricts the polarization values to a small number of distinct groups. There groups in turn are topologically inequivalent, making polarization a topological index. We also conduct density functional theory calculations to obtain actual polarization values of materials belonging to C3 symmetry and show that our results are consistent with our theoretical analysis. Finally we prove that any transformation from one class of polarization to another is a topological phase transition. In Chapter 5 we use density functional theory to examine the electronic properties of graphene intercalation compounds. Bilayer pseudospin field effect transistor (BiSFET) has been proposed as an interesting low-power, efficient post-CMOS switch. In order to implement this device we need bilayer graphene with reduced interlayer interaction. One way of achieving that is by inserting foreign molecules between the layers, a process which is called intercalation. In this chapter we examine the electronic properties of bilayer graphene intercalated with iodine monochloride and iodine monobromide molecules. We find that intercalation of graphene indeed makes it promising for the implementation of BiSFET, by reducing interlayer interaction. As an interesting side problem, we also use hybrid, more extensive approaches in DFT, to examine the electronic and optical properties of dilute nitrides. Dilute nitrides are highly promising and interesting materials for the purposes of optoelectronic applications. Together, we hope this work helps in elucidating the electronic properties of promising material systems as well as act as a guide for experimentalists. / text

A critical set in a Latin square of order n is a set of entries in an n×n array which can be embedded in precisely one Latin square of order n, with the property that if any entry of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order n. The number of critical sets grows super-exponentially as the order of the Latin square increases. It is difficult to find patterns in Latin squares of small order (order 5 or less) which can be generalised in the process of creating new theorems. Thus, I have written many algorithms to find critical sets with various properties in Latin squares of order greater than 5, and to deal with other related structures. Some algorithms used in the body of the thesis are presented in Chapter 3; results which arise from the computational studies and observations of the patterns and subsequent results are presented in Chapters 4, 5, 6, 7 and 8. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978 Curran and van Rees proved that lcs(n)<=n²-n. In Chapter 4, it is shown that lcs(n)<=n²-3n+3. Chapter 5 provides new bounds on the maximum number of intercalates in Latin squares of orders m×2^α (m odd, α>=2) and m×2^α+1 (m odd, α>=2 and α≠3), and a new lower bound on lcs(4m). It also discusses critical sets in intercalate-rich Latin squares of orders 11 and 14. In Chapter 6 a construction is given which verifies the existence of a critical set of size n²÷ 4 + 1 when n is even and n>=6. The construction is based on the discovery of a critical set of size 17 for a Latin square of order 8. In Chapter 7 the representation of Steiner trades of volume less than or equal to nine is examined. Computational results are used to identify those trades for which the associated partial Latin square can be decomposed into six disjoint Latin interchanges. Chapter 8 focusses on critical sets in Latin squares of order at most six and extensive computational routines are used to identify all the critical sets of different sizes in these Latin squares.

A critical set in a Latin square of order n is a set of entries in an n×n array which can be embedded in precisely one Latin square of order n, with the property that if any entry of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order n. The number of critical sets grows super-exponentially as the order of the Latin square increases. It is difficult to find patterns in Latin squares of small order (order 5 or less) which can be generalised in the process of creating new theorems. Thus, I have written many algorithms to find critical sets with various properties in Latin squares of order greater than 5, and to deal with other related structures. Some algorithms used in the body of the thesis are presented in Chapter 3; results which arise from the computational studies and observations of the patterns and subsequent results are presented in Chapters 4, 5, 6, 7 and 8. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978 Curran and van Rees proved that lcs(n)<=n²-n. In Chapter 4, it is shown that lcs(n)<=n²-3n+3. Chapter 5 provides new bounds on the maximum number of intercalates in Latin squares of orders m×2^α (m odd, α>=2) and m×2^α+1 (m odd, α>=2 and α≠3), and a new lower bound on lcs(4m). It also discusses critical sets in intercalate-rich Latin squares of orders 11 and 14. In Chapter 6 a construction is given which verifies the existence of a critical set of size n²÷ 4 + 1 when n is even and n>=6. The construction is based on the discovery of a critical set of size 17 for a Latin square of order 8. In Chapter 7 the representation of Steiner trades of volume less than or equal to nine is examined. Computational results are used to identify those trades for which the associated partial Latin square can be decomposed into six disjoint Latin interchanges. Chapter 8 focusses on critical sets in Latin squares of order at most six and extensive computational routines are used to identify all the critical sets of different sizes in these Latin squares.

A critical set in a Latin square of order n is a set of entries in an n×n array which can be embedded in precisely one Latin square of order n, with the property that if any entry of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order n. The number of critical sets grows super-exponentially as the order of the Latin square increases. It is difficult to find patterns in Latin squares of small order (order 5 or less) which can be generalised in the process of creating new theorems. Thus, I have written many algorithms to find critical sets with various properties in Latin squares of order greater than 5, and to deal with other related structures. Some algorithms used in the body of the thesis are presented in Chapter 3; results which arise from the computational studies and observations of the patterns and subsequent results are presented in Chapters 4, 5, 6, 7 and 8. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978 Curran and van Rees proved that lcs(n)<=n²-n. In Chapter 4, it is shown that lcs(n)<=n²-3n+3. Chapter 5 provides new bounds on the maximum number of intercalates in Latin squares of orders m×2^α (m odd, α>=2) and m×2^α+1 (m odd, α>=2 and α≠3), and a new lower bound on lcs(4m). It also discusses critical sets in intercalate-rich Latin squares of orders 11 and 14. In Chapter 6 a construction is given which verifies the existence of a critical set of size n²÷ 4 + 1 when n is even and n>=6. The construction is based on the discovery of a critical set of size 17 for a Latin square of order 8. In Chapter 7 the representation of Steiner trades of volume less than or equal to nine is examined. Computational results are used to identify those trades for which the associated partial Latin square can be decomposed into six disjoint Latin interchanges. Chapter 8 focusses on critical sets in Latin squares of order at most six and extensive computational routines are used to identify all the critical sets of different sizes in these Latin squares.