Prime Rational Functions and Integral Polynomials

Let f(x) be a complex rational function. In this work, we study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we derive some conditions for the case of complex polynomials. We consider also the divisibility of integral polynomials, and we present a generalization of a theorem of Nieto. We show that if f(x) and g(x) are integral polynomials such that the content of g divides the content of f and g(n) divides f(n) for an integer n whose absolute value is larger than a certain bound, then g(x) divides f(x) in Z[x]. In addition, given an integral polynomial f(x), we provide a method to determine if f is irreducible over Z, and if not, find one of its divisors in Z[x].

Scales, networks and uncertainty : an examination of environmental policy-making in Ontario

Through a case-study analysis of Ontario's ethanol policy, this thesis addresses a number of themes that are consequential to policy and policy-making: spatiality, democracy and uncertainty. First, I address the 'spatial debate' in Geography pertaining to the relevance and affordances of a 'scalar' versus a 'flat' ontoepistemology. I argue that policy is guided by prior arrangements, but is by no means inevitable or predetermined. As such, scale and network are pragmatic geographical concepts that can effectively address the issue of the spatiality of policy and policy-making. Second, I discuss the democratic nature of policy-making in Ontario through an examination of the spaces of engagement that facilitate deliberative democracy. I analyze to what extent these spaces fit into Ontario's environmental policy-making process, and to what extent they were used by various stakeholders. Last, I take seriously the fact that uncertainty and unavoidable injustice are central to policy, and examine the ways in which this uncertainty shaped the specifics of Ontario's ethanol policy. Ultimately, this thesis is an exercise in understanding sub-national environmental policy-making in Canada, with an emphasis on how policy-makers tackle the issues they are faced with in the context of environmental change, political-economic integration, local priorities, individual goals, and irreducible uncertainty.

Geochemistry and petrogenesis of the McElroy and Larder Lake assemblages, Abitibi Greenstone Belt, Northeastern Ontario

The McElroy and Larder Lake assemblages, located in the southern Abitibi Greenstone Belt are two late Archean metavolcanic sequences having markedly contrasting physical characteristics arid are separated from one another by a regional fault. An assemblage is an informal term which describes stratified volcanic and/or sedimentary rock units built during a specific time period in a similar depositional or volcanic setting and are commonly bounded by faults, unconformities or intrusions. The petrology and petrogenesis of these assemblages have been investigated to determine if a genetic link exists between the two adjacent assemblages. The McElroy assemblage is homoclinal sequence of evolved massive and pillowed fl.ows, which except for the basal unit represents a progressively fractionated volcanic pile. From the base to the top of the assemblage the lithologies include Fe-tholeiitic, dendritic flows; komatiite basaltic, ultramafic flows; Mg-tholeiitic, leucogabbro; Mg-tholeiitic, massive flows and Fe-tholeiitic, pillowed flows. Massive flows range from coarse grained to aphanitic and are commonly plagioclase glomerophyric. The Larder Lake assemblage consists of komatiitic, Mg-rich and Fe-rich tholeiitic basalts, structurally disrupted by folds and faults. Tholeiitic rocks in the Larder Lake assemblage range from aphanitic to coarse grained massive and pillowed flows. Komatiitic flows contain both spinifex and massive textures. Geochemical variability within both assemblages is attributed to different petrogenetic histories. The lithologies of the McElroy assemblage were derived by partial melting of a primitive mantle source followed by various degrees of crystal fractionation. Partial melting of a primitive mantle source generated the ultramafic flows and possibly other flows in the assemblage. Fractionation of ultramafic flows may have also produced the more evolved McElroy lithologies. The highly evolved, basal, dendritic flow may represent the upper unit 3 of a missing volcanic pile in which continued magmatism generated the remaining McElroy lithologies. Alternatively, the dendritic flows may represent a primary lava derived from a low degree (10-15%) partial melt of a primitive mantle source which was followed by continued partial melting to generate the ultramafic flows. The Larder Lake lithologies were derived by partial melting of a komatiitic source followed by gabbroic fractionation. The tectonic environment for both assemblages is interpreted to be an oceanic arc setting. The McElroy assemblage lavas were generated in a mature back arc setting whereas the Larder Lake lithologies were produced during the early stages of komatiitc crust subduction. This setting is consistent with previous models involving plate tectonic processes for the generation of other metavolcanic assemblages in the Abitibi Greenstone Belt.

Collective violence : a study of the gendered and socio-economic factors behind early modern Italian and English witch hunts

In this study, I build upon my previous research in which I focus on religious doctrine as a gendered disciplinary apparatus, and examine the witch trials in early modem England and Italy in light of socio-economic issues relating to gender and class. This project examines the witch hunts/trials and early modem visual representations of witches, and what I suggest is an attempt to create docile bodies out of members of society who are deemed unruly, problematic and otherwise 'undesirable'; it is the witch's body that is deemed counternormative. This study demonstrates that it is neighbours and other acquaintances of accused witches that take on the role of the invisible guard of Bantham's Panoptic on. As someone who is trained in the study of English literature and literary theory, my approach is one that is informed by this methodology. It is my specialization in early modem British literature that first exposed me to witch-hunting manuals and tales of the supernatural, and it is for this reason that my research commences with a study of representations of witches and witchcraft in early modem England. From my initial exposure to such materials I proceed to examine the similarities and the differences of the cultural significance of the supernatural vis-a.-vis women's activities in early modem Italy. The subsequent discussion of visual representations of witches involves a predominance of Germanic artists, as the seminal work on the discernment of witches and the application of punishment known as the Malleus Meleficarum, was written in Germany circa 1486. Textual accounts of witch trials such as: "A Pitiless Mother (1616)," "The Wonderful Discovery of the Witchcrafts of Margaret and Philippa Flower (1619)," "Magic and Poison: The Trial ofChiaretta and Fedele (circa 1550)", and the "The Case of Benvegnuda Pincinella: Medicine Woman or Witch (1518),"and witchhunting manuals such as the Malleus Melejicarum and Strix will be put in direct dialogue with visual representations of witches in light of historical discourses pertaining to gender performance and gendered expectations. Issues relating to class will be examined as they pertain to the material conditions of presumed witches. The dominant group in any temporal or geographic location possesses the tools of representation. Therefore, it is not surprising that the physical characteristics, sexual habits and social material conditions that are attributed to suspected witches are attributes that can be deemed deviant by the ruling class. The research will juxtapose the social material conditions of suspected witches with the guilt, anxiety, and projection of fear that the dominant groups experienced in light of the changing economic landscape of the Renaissance. The shift from feudalism to primitive accumulation, and capitalism saw a rise in people living in poverty and therefore an increased dependence upon the good will of others. I will discuss the social material conditions of accused witches as informed by what Robyn Wiegman terms a "minoritizing discourse" (210). People of higher economic standing often blamed their social, medical, and/or economic difficulties on the less fortunate, resulting in accusations of witchcraft.

Region Connection Calculus: Composition Tables and Constraint Satisfaction Problems

Qualitative spatial reasoning (QSR) is an important field of AI that deals with qualitative aspects of spatial entities. Regions and their relationships are described in qualitative terms instead of numerical values. This approach models human based reasoning about such entities closer than other approaches. Any relationships between regions that we encounter in our daily life situations are normally formulated in natural language. For example, one can outline one's room plan to an expert by indicating which rooms should be connected to each other. Mereotopology as an area of QSR combines mereology, topology and algebraic methods. As mereotopology plays an important role in region based theories of space, our focus is on one of the most widely referenced formalisms for QSR, the region connection calculus (RCC). RCC is a first order theory based on a primitive connectedness relation, which is a binary symmetric relation satisfying some additional properties. By using this relation we can define a set of basic binary relations which have the property of being jointly exhaustive and pairwise disjoint (JEPD), which means that between any two spatial entities exactly one of the basic relations hold. Basic reasoning can now be done by using the composition operation on relations whose results are stored in a composition table. Relation algebras (RAs) have become a main entity for spatial reasoning in the area of QSR. These algebras are based on equational reasoning which can be used to derive further relations between regions in a certain situation. Any of those algebras describe the relation between regions up to a certain degree of detail. In this thesis we will use the method of splitting atoms in a RA in order to reproduce known algebras such as RCC15 and RCC25 systematically and to generate new algebras, and hence a more detailed description of regions, beyond RCC25.

Phonon spectra and temperature variation of thermodynamic properties of fcc metals via Finnis-Sinclair type many body potentials: Sutton-Chen and improved Sutton-Chen models

Volume(density)-independent pair-potentials cannot describe metallic cohesion adequately as the presence of the free electron gas renders the total energy strongly dependent on the electron density. The embedded atom method (EAM) addresses this issue by replacing part of the total energy with an explicitly density-dependent term called the embedding function. Finnis and Sinclair proposed a model where the embedding function is taken to be proportional to the square root of the electron density. Models of this type are known as Finnis-Sinclair many body potentials. In this work we study a particular parametrization of the Finnis-Sinclair type potential, called the "Sutton-Chen" model, and a later version, called the "Quantum Sutton-Chen" model, to study the phonon spectra and the temperature variation thermodynamic properties of fcc metals. Both models give poor results for thermal expansion, which can be traced to rapid softening of transverse phonon frequencies with increasing lattice parameter. We identify the power law decay of the electron density with distance assumed by the model as the main cause of this behaviour and show that an exponentially decaying form of charge density improves the results significantly. Results for Sutton-Chen and our improved version of Sutton-Chen models are compared for four fcc metals: Cu, Ag, Au and Pt. The calculated properties are the phonon spectra, thermal expansion coefficient, isobaric heat capacity, adiabatic and isothermal bulk moduli, atomic root-mean-square displacement and Gr\"{u}neisen parameter. For the sake of comparison we have also considered two other models where the distance-dependence of the charge density is an exponential multiplied by polynomials. None of these models exhibits the instability against thermal expansion (premature melting) as shown by the Sutton-Chen model. We also present results obtained via pure pair potential models, in order to identify advantages and disadvantages of methods used to obtain the parameters of these potentials.