This paper explores whether Cartesian dualism prioritizes the masculine over the feminine. Feminist authors have argued that due to the prioritization of the mind over the body in Cartesian dualism and the association of the masculine with the mind and the association of the feminine with the body, the masculine is prioritized. This paper analyzes both this prioritization of the mind over the body and the association of the masculine with the mind and the feminine with the body.

This work details a multigrid-accelerated cut-cell Cartesian mesh methodology for the solution of a single partial differential equation set that describes incompressible as well as compressible flow. The latter includes sub-, trans- and supersonic flows. Cut-cell technology is developed which furnishes body-fitted meshes with an overlapping Cartesian mesh as starting point, and in a manner which is insensitive to surface definition inconsistencies. An edge-based vertex-centred finite volume method is employed for the purpose of spatial discretisation. Further, an alternative dual-mesh construction strategy is developed and the standard discretisation scheme suitably enhanced. Incompressibility is dealt with via a locally preconditioned artificial compressibility algorithm, and stabilisation is in all cases achieved with scalar-valued artificial dissipation. In transonic flows, shocks are captured via pressure switch-activated upwinding. The solution process is accelerated by the use of a full approximation scheme (FAS) multigrid method where coarse meshes are generated automatically via a volume agglomeration methodology. The developed modelling technology is validated by application to the solution of a number of benchmark problems. The standard discretisation as well as the alternative method are found to be equivalent in terms of both accuracy and computational cost. Finally, the multigrid implementation is shown to achieve decreases in CPU time of between a factor two to one order of magnitude. In the context of cut-cell Cartesian meshes, the above work has resulted in the following novel contributions: the development of an alternative vertex-centred discretisation method; the use of volume agglomerated multigrid solution technology and the use of a single equation set for both incompressible and compressible flows. / Dissertation (MEng (Mechanical Engineering))--University of Pretoria, 2007. / Mechanical and Aeronautical Engineering / unrestricted

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This dissertation discusses the life and the writings of the seventeenth-century Dutch Cartesian Johannes Swartenhengst (1644-1711). Thus far Swartenhengst has always been an obscure and little-noticed figure in the history of the Early Dutch Enlightenment, who is only briefly mentioned in a couple of secondary sources due to his intellectual association with the Flemish philosopher Arnold Geulincx (1624-69). In recent years I have discovered fourteen previously unknown disputations that Swartenhengst presided over during his career as lector at Leiden in the early 1670s. Swartenhengst’s appointment at this university was, however, soon terminated on account of his overzealous defence of Cartesian philosophy, and no significant details have remained from his life hereafter. Although Swartenhengst’s disputations bear a somewhat concise and impersonal character (as is typical for the genre), they touch upon all the major philosophical disciplines that were then taught at the university. Swartenhengst’s dismissal occurred at a particularly heated moment, when the ecclesiastical pressure that had been building up since the political changes of 1672 now finally culminated at the university. His disputations, therefore, provide us with an interesting example of the Cartesian views that were circulating in academic circles, but which were apparently no longer tolerated. More importantly, however, Swartenhengst’s disputations also provide us with an interesting case study of the immediate continuation of Geulincx’s philosophy at Leiden, whose views soon disappeared into oblivion on account of their association with Spinozism during the early eighteenth century. Apart from offering a detailed account of Swartenhengst’s biographical details and a discussion of the major theological problems that were associated with René Descartes’ philosophy, this dissertation also includes an analysis of the content of his disputations, which focuses on the topics of occasionalism, epistemology, and natural law. Finally it will be asked how closely Swartenhengst’s disputations related to the views of his teacher Arnold Geulincx; and whether he should be labelled a ‘radical Cartesian’ on account of the content of his teaching? Although Swartenhengst was only a minor player in the history of the Early Dutch Enlightenment, the details of his life and writings certainly represent a unique and interesting story, which can also contribute to our general understanding of the period.

Chomsky’s Cartesian Linguistics frames into his linguistic work and the resulting debates between rationalists and empiricists. I focus on the following key aspects: (i) the historic connection of Cartesian Linguistics to previous linguistic theorizing, (ii) the development of Chomsky’s own theorizing, (iii) the empirical work addressing the problem of language acquisition and (iv) the problem of computational modeling of language learning. Chomsky claims that his view is situated within a rationalist Cartesian tradition and that only rationalists will be able to account fully for all aspects of human language. My thesis challenges both claims. I found only remote connections between Cartesian and Chomskyan commitments. Chomsky holds that (i) language is species-specific, (ii) language is domain-specific, and (iii) language acquisition depends on innate knowledge. Descartes accepted (i), but argued that language is an indicator of domain-general intelligence. Innate resources play a different role for language acquisition for Chomsky and for Descartes. Chomsky revived linguistics during the 1950s by promising to make it a rigorous part of the biological sciences. However, his work has not resulted in a better understanding of language acquisition and use. Key concepts like ‘innateness’, ‘Universal Grammar’ and ‘Language Acquisition Device’ remain in need of precise definition, and the Poverty of the Stimulus Argument does not rule out data-driven domain-general language acquisition. Empirical work in developmental psychology has demonstrated that children acquire and practice many language-related cognitive abilities long before they produce their first words. Chomsky’s dictum that language learning is uniform across the species and invariably follows genetically determined stages remains empirically unconfirmed. Computational modeling has accounted for some internal structure of language acquisition mechanisms and simulates the specific conditions under which children learn language. Contemporary models use samples of child-directed-speech as input and have replicated numerous aspects of human performance. Given my findings I suggest that Chomskyan linguistics is not Cartesian in substance or in spirit. Descartes was wary of those “who take no account of experience and think that truth will spring from their brains like Minerva from the head of Jupiter” (CSM I, p. 21). His science relied on sense experience (empiricism) and deduction (rationalism) and a truly Cartesian Linguistics will revive this part of the Cartesian tradition.

Let $G\ensuremath{\mathbin{\raisebox{0.3mm}{$\scriptstyle\square$}}} H$ denote the Cartesian product of the graphs $G$ and $H$. Domination of the Cartesian product of two graphs has received much attention, with a main objective to confirm the truth of Vizing's well-known conjecture. The conjecture states that the domination number of the Cartesian product of two graphs is at least as large as the product of the respective domination numbers. The potential truth of Vizing's conjecture gives rise to investigating the domination of graph products that generalizes the Cartesian product. The generalized prism $\pi G$ of $G$ is the graph consisting of two copies of $G$, with edges between the copies determined by a permutation $\pi$ acting on the vertices of $G$. A generalized Cartesian product $G\ensuremath{\mathbin{\raisebox{0.3mm}{${\scriptstyle \square}$}\hspace{-1.99mm}\raisebox{0.65mm}{${\scriptstyle \pi}$}}} H$ is defined here, incorporating structural properties of both the Cartesian product of two graphs as well as the generalized prism of a graph. Conditions on the isomorphism of two generalized Cartesian products are explored first, establishing a characterization in the case of natural isomorphisms. A comparison of the diameter of the generalized Cartesian product and the corresponding Cartesian product graph is used to illustrate the structural differences between these graph products. This comparison is continued through a study of the validity of an inequality similar to Vizing's conjecture for Cartesian products. Graphs that attain equality in the general bounds for the domination number of the Cartesian product and generalized Cartesian product are investigated in more detail. For any graph $G$ and $n\geq 2$, $\min\{|V(G)|,\gamma(G)+n-2\}\leq\gamma(G\ensuremath{\mathbin{\raisebox{0.3mm}{$\scriptstyle\square$}}} K_{n})\leq n\gamma(G)$. A graph $G$ is called a consistent Cartesian fixer if $\gamma(G\ensuremath{\mathbin{\raisebox{0.3mm}{$\scriptstyle\square$}}} K_{n})=\gamma(G)+n-2$ for each $n$ such that $2\leq n<|V(G)|-\gamma(G)+2$. A graph attaining equality in the stated upper bound on $\gamma(G\ensuremath{\mathbin{\raisebox{0.3mm}{$\scriptstyle\square$}}} K_{n})$ is called a Cartesian $n$-multiplier. Both of these classes are characterized. Concerning the generalized Cartesian product, $\gamma(G\ensuremath{\mathbin{\raisebox{0.3mm}{${\scriptstyle \square}$}\hspace{-1.99mm}\raisebox{0.65mm}{${\scriptstyle \pi}$}}} K_{n})\leq n\gamma(G)$ for any graph $G$, permutation $\pi$ and $n\geq 2$. A graph attaining equality in the upper bound for all $\pi$ is called a universal multiplier. Such graphs are characterized similar to a known result for generalized prisms. A similar problem for the product $G\ensuremath{\mathbin{\raisebox{0.3mm}{${\scriptstyle \square}$}\hspace{-1.99mm}\raisebox{0.65mm}{${\scriptstyle \pi}$}}} C_{n}$ is considered, with conditions on a graph being a so-called cycle multiplier provided. A graph attaining equality in the lower bound $\gamma(G\ensuremath{\mathbin{\raisebox{0.3mm}{${\scriptstyle \square}$}\hspace{-1.99mm}\raisebox{0.65mm}{${\scriptstyle \pi}$}}} H)\geq\gamma(G)$ for some permutation $\pi$ is called a $\pi$-$H$-fixer. A brief investigation is conducted into the existence of universal $H$-fixers, i.e.~graphs that are $\pi$-$H$-fixers for some $H$ and all permuations $\pi$ of $V(G)$, and it is shown that no such graphs exist when $n\geq 3$. A known efficient algorithm for determining $\gamma(G\ensuremath{\mathbin{\raisebox{0.3mm}{$\scriptstyle\square$}}} P_{n})$ is surveyed, and modified to accommodate any Cartesian product $G\ensuremath{\mathbin{\raisebox{0.3mm}{$\scriptstyle\square$}}} H$, thereby establishing a general framework for evaluating the domination number of $G\ensuremath{\mathbin{\raisebox{0.3mm}{$\scriptstyle\square$}}} H$ for a fixed graph $G$ and any $H$. An algorithm to determine $\gamma(G\ensuremath{\mathbin{\raisebox{0.3mm}{$\scriptstyle\square$}}} T)$ for any tree $T$ is provided, and it is observed to be polynomial for trees of bounded maximum degree. The general framework for $G\ensuremath{\mathbin{\raisebox{0.3mm}{$\scriptstyle\square$}}} H$ is also modified to accommodate the generalized Cartesian product $G\ensuremath{\mathbin{\raisebox{0.3mm}{${\scriptstyle \square}$}\hspace{-1.99mm}\raisebox{0.65mm}{${\scriptstyle \pi}$}}} H$. The study diverts from the main topic of domination to investigate the planarity of the generalized Cartesian product graph. If both $G$ and $H$ are 2-connected graphs, then $G\ensuremath{\mathbin{\raisebox{0.3mm}{${\scriptstyle \square}$}\hspace{-1.99mm}\raisebox{0.65mm}{${\scriptstyle \pi}$}}} H$ is nonplanar. A known simple polynomial-time planarity testing algorithm is surveyed, and used to establish conditions on the planarity of $P_{m}\ensuremath{\mathbin{\raisebox{0.3mm}{${\scriptstyle \square}$}\hspace{-1.99mm}\raisebox{0.65mm}{${\scriptstyle \pi}$}}} P_{n}$, the generalized Cartesian product of two paths. This research aims to lay the foundation on which further properties of the generalized Cartesian product and further generalizations may be studied, as well as to provide various open problems to spark interest in the research area.

The aim of this thesis is to investigate the passions in Cartesian philosophy. It analyses the nature, characteristics and the causes of passions as discussed by Descartes in his correspondence with Princess Elizabeth and his last book The Passions of the Soul (Les passions de l&rsquo / &acirc / me). This thesis purports to explain Descartes&rsquo / ethical view by examining the physical mechanism of the passions and their relation to the soul. The reason, will and their essential roles in Cartesian ethics are discussed.

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In this thesis, we analyze theoretical properties of the discontinuous Galerkin method (DGM) and propose novel approaches to implementation with the aim to increase its efficiency. First, we derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal [p/p+1] Pade approximation of exp(-z) when the p-th degree basis functions are used. Then, we extend the analysis to nonuniform meshes where both the size of elements and the composition of the mesh influence the spectrum. We show that the spectrum depends on the ratio of the size of the largest to the smallest cell as well as the number of cells of different types. We find that the spectrum grows linearly as a function of the proportion of small cells present in the mesh when the size of small cells is greater than some critical value. When the smallest cells are smaller than this critical value, the corresponding eigenvalues lie outside of the main spectral curve. Numerical examples on nonuniform meshes are presented to show the improvement on the time step restriction. In particular, this result can be used to improve the time step restriction on Cartesian grids. Finally, we present a discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries. Cutting an embedded geometry out of the Cartesian grid creates cut cells, which are difficult to deal with for two reasons. One is the restrictive CFL number and the other is the integration on irregularly shaped cells. We use explicit time integration employing cell merging to avoid restrictively small time steps. We provide an algorithm for splitting complex cells into triangles and use standard quadrature rules on these for numerical integration. To avoid the loss of accuracy due to straight sided grids, we employ the curvature boundary conditions. We show that the proposed method is robust and high-order accurate.

We consider computation of resonances in open systems and acoustic scattering problems. These problems are posed on an unbounded domain and domain truncation is required for the numerical computation. In this paper, a perfectly matched layer (PML) technique is proposed for computation of solutions to the unbounded domain problems. For resonance problems, resonance functions are characterized as improper eigenfunction (non-zero solutions of the eigenvalue problem which are not square integrable) of the Helmholtz equation on an unbounded domain. We shall see that the application of the spherical PML converts the resonance problem to a standard eigenvalue problem on the infinite domain. Then, the goal will be to approximate the eigenvalues first by replacing the infinite domain by a finite computational domain with a convenient boundary condition and second by applying finite elements to the truncated problem. As approximation of eigenvalues of problems on a bounded domain is classical [12], we will focus on the convergence of eigenvalues of the (continuous) PML truncated problem to those of the infinite PML problem. Also, it will be shown that the domain truncation does not produce spurious eigenvalues provided that the size of computational domain is sufficiently large. The spherical PML technique has been successfully applied for approximation of scattered waves [13]. We develop an analysis for the case of a Cartesian PML application to the acoustic scattering problem, i.e., solvability of infinite and truncated Cartesian PML scattering problems and convergence of the truncated Cartesian PML problem to the solution of the original solution in the physical region as the size of computational domain increases.