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This thesis investigates the synthesis and structural characterisation of AM(_2)O(_8) phases, many of which show negative thermal expansion (NTE); relevant literature is reviewed in Chapter One. Chapter Two describes the synthesis, structure solution, and mechanistic role of a new family of low-temperature (LT) orthorhombic AM(_2)O(_8) polymorphs (A(^TV) = Zr, Hf; M(^VI) = Mo, W). These materials are key intermediates in the preparation of cubic AM(_2)O(_8) phases from AM(_2)O(_7)(OH)(_2)(H(_2)O)(_2). The structure of LT-AM(_2)O(_8) has been elucidated by combined laboratory X-ray and neutron powder diffraction. Variable temperature X-ray diffraction (VTXRD) studies have shown LT- AM(_2)O(_8) phases exhibit anisotropic NTE. LT-ZrMo(_2)O(_8) has been shown to undergo spontaneous rehydration, allowing preparation of ZrMo(_2)O(_7)(OD)(_2)(D(_2)O)(_2) and assignment of D(_2)O/OD positions within the structure by neutron diffraction. Using this result, a reversible topotactic dehydration pathway from AM(_2)O(_7)(OH)(_2)(H(_2)O)(_2) to LT-AM(_2)O(_8)s is proposed. Chapter Three investigates the order-disorder phase transition with concurrent oxygen mobility in cubic AM(_2)O(_8) materials; studies include comprehensive VT neutron diffraction of cubic ZrMo(_2)O(_8) to reveal a static to dynamic transition at 215 K, and novel quench-anneal/quench-warm variable temperature/time diffraction experiments on ZrMo(_2)O(_8) which lead to an activation energy of 40 kJmol(^-1) for oxygen migration. In Chapter Four (^17)O-labelled cubic ZrW(_2)O(_8) has been prepared to understand the oxygen migration process by VT MAS NMR. In situ hydrothermal studies of cubicZrMo(_2)O(_8) using synchrotron radiation have shown direct hydration to ZrMo(_2)O(_7)(OH)(_2)(H(_2)O)(_2).. In Chapter Five VTXRD of trigonal a-AMo(_2)O(_8) phases reveals a previously unknown second-order phase transition at 487 K (A = Zr) or 463 K (A = Hf) from P31c to P3ml. Rigid-body Rietveld refinements have shown this is due to alignment of apical Mo-O groups with the c axis in the high-temperature, a' phase.

Nonnegative matrices have a myriad of applications in the biological, social, and physical genres. Of particular importance are the primitive matrices. A nonnegative matrix, M, is primitive exactly when there is a positive integer, k, such that M[superscript k] has only positive entries; that is, all the entries in M[superscript k] are strictly greater than zero. This method of determining if a matrix is primitive uses matrix multiplication and so would require time ���(n[superscipt ��]) where ��>2.3 even if fast matrix multiplication were used. Our goal is to find a much faster algorithm. This can be achieved by viewing a nonnegative matrix, M, as the adjacency matrix for a graph, G(M). The matrix, M, is primitive if and only if G(M) is strongly connected and the greatest common divisor of the cycle lengths in G(M) is 1. We devised an algorithm based in breadth-first search which finds a set of cycle lengths whose gcd is the same as that of G(M). This algorithm has runtime O(e) where e is the number of nonzero entries in M and therefore equivalent to the number of edges in G(M). A proof is given shown the runtime of O(n + e) along with some empirical evidence that supports this finding. / Graduation date: 2003

Positivity of polynomials, as a key notion in real algebra, is one of the oldest topics. In a given context, some polynomials can be represented in a form that reveals their positivity immediately, like sums of squares. A large body of literature deals with the question which positive polynomials can be represented in such a way.<p>The milestone in this development was Schm"udgen's solution of the moment problem for compact semi-algebraic sets. In 1991, Schm"udgen proved that if the associated basic closed semi-algebraic set $K_{S}$ is compact, then any polynomial which is strictly positive on $K_{S}$ is contained in the preordering $T_{S}$.<p>Putinar considered a further question: when are `linear representations' possible? He provided the first step in answering this question himself in 1993. Putinar proved if the quadratic module $M_{S}$ is archimedean, any polynomial which is strictly positive on $K_{S}$ is contained in $M_{S}$, i.e., has a linear representation.<p>In the present thesis, we concentrate on the linear representations in the one variable polynomial ring. We first investigate the relationship of the two conditions in Schm"udgen's Theorem and Putinar's Criterion: $K_{S}$ compact and $M_{S}$ archimedean. They are actually equivalent. We find another proof for this result and hereby we can improve Schm"udgen's Theorem in the one variable case.<p>Secondly, we investigate the relationship of $M_{S}$ and $T_{S}$. We use elementary arguments to prove in the one variable case when $K_{S}$ is compact, they are equal.<p>Thirdly, we present Scheiderer's Main Theorem with a detailed proof. Scheiderer established a local-global principle for the polynomials non-negative on $K_{S}$ to be contained in $M_{S}$ in 2003. This principle which we call Scheiderer's Main Theorem here extends Putinar's Criterion.<p>Finally, we consider Scheiderer's Main Theorem in the one variable case, and give a simplified version of this theorem. We also apply this Simple Version of the Main Theorem to give some elementary proofs for existing results.

Positivity of polynomials, as a key notion in real algebra, is one of the oldest topics. In a given context, some polynomials can be represented in a form that reveals their positivity immediately, like sums of squares. A large body of literature deals with the question which positive polynomials can be represented in such a way.<p>The milestone in this development was Schm"udgen's solution of the moment problem for compact semi-algebraic sets. In 1991, Schm"udgen proved that if the associated basic closed semi-algebraic set $K_{S}$ is compact, then any polynomial which is strictly positive on $K_{S}$ is contained in the preordering $T_{S}$.<p>Putinar considered a further question: when are `linear representations' possible? He provided the first step in answering this question himself in 1993. Putinar proved if the quadratic module $M_{S}$ is archimedean, any polynomial which is strictly positive on $K_{S}$ is contained in $M_{S}$, i.e., has a linear representation.<p>In the present thesis, we concentrate on the linear representations in the one variable polynomial ring. We first investigate the relationship of the two conditions in Schm"udgen's Theorem and Putinar's Criterion: $K_{S}$ compact and $M_{S}$ archimedean. They are actually equivalent. We find another proof for this result and hereby we can improve Schm"udgen's Theorem in the one variable case.<p>Secondly, we investigate the relationship of $M_{S}$ and $T_{S}$. We use elementary arguments to prove in the one variable case when $K_{S}$ is compact, they are equal.<p>Thirdly, we present Scheiderer's Main Theorem with a detailed proof. Scheiderer established a local-global principle for the polynomials non-negative on $K_{S}$ to be contained in $M_{S}$ in 2003. This principle which we call Scheiderer's Main Theorem here extends Putinar's Criterion.<p>Finally, we consider Scheiderer's Main Theorem in the one variable case, and give a simplified version of this theorem. We also apply this Simple Version of the Main Theorem to give some elementary proofs for existing results.