A logic that utilizes higher-order quantification --quantifying over concepts (or relations), not just over the first-order level of individuals-- can be interpreted standardly or nonstandardly depending on whether one takes an intensional or extensional view of concepts. I argue that this decision is connected to how one understands the mathematical notion of function. A function is often understood as a rule that, when given an argument from a set of objects called a "domain," returns a value from a set of objects called a "codomain." Because a concept can be thought of as a two-valued function (that indicates whether or not a given object falls under the concept), having an extensional interpretation of higher-order variables --the standard interpretation-- requires that one adopt an extensional notion of function. Viewed extensionally, however, a function is understood not as a rule but rather as a correlation associating every element in a domain with an element in a codomain. When the domain is finite, the two understandings of function are equivalent (since one can define a rule for any finite correlation), but with an infinite domain, the latter understanding admits arbitrary functions, or correlations not definable by a finitely specifiable rule. Rejection of the standard interpretation is often motivated by the same reasons used to resist the extensional understanding of function. Such resistance is overt in the pronouncements of Leopold Kronecker, but is also implicit in the work of Gottlob Frege, who used an intensional notion of function in his logic. Looking at the problem historically, I argue that the extensional notion of function has been basic to mathematics since ancient times. Moreover, I claim that Gottfried Wilhelm Leibniz's combination of mathematical and metaphysical ideas helped inaugurate an extensional and ultimately model-theoretical approach to mathematical concepts that led to some of the most important applications of mathematics to science (e.g. the use of non-Euclidean geometry in the theory of general relativity). In logic, Frege's use of an intensional notion of function led to contradiction, while Richard Dedekind and Georg Cantor applied the extensional notion of function to develop mathematically revolutionary theories of the transfinite. / 2025-10-15

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第18402号 / 情博第517号 / 新制||情||91(附属図書館) / 31260 / 京都大学大学院情報学研究科複雑系科学専攻 / (主査)教授 山本 裕, 教授 西村 直志, 教授 太田 快人 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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The problem of finding a reduced order model, optimal in the H² sense, to a given system model is a fundamental one in control system analysis and design. The addition of a H^∞ constraint to the H² optimal model reduction problem results in a more practical yet computationally more difficult problem. Without the global convergence of homotopy methods, both the H² optimal and the combined H²/H^∞ model reduction problems are very difficult. For both problems homotopy algorithms based on several formulations—input normal form; Ly, Bryson, and Cannon's 2 X 2 block parametrization; a new nonminimal parametrization—are developed and compared here. For the H² optimal model order reduction problem, these numerical algorithms are also compared with that based on Hyland and Bernstein's optimal projection equations. Both the input normal form and Ly form are very efficient compared to the over parametrization formulation and the optimal projection equations approach, since they utilize the minimal number of possible degrees of freedom. However, they can fail to exist or be very ill conditioned. The conditions under which the input normal form and the Ly form become ill conditioned are examined. The over-parametrization formulation solves the ill conditioning issue, and usually is more efficient than the approach based on solving the optimal projection equations for the H² optimal model reduction problem. However, the over-parametrization formulation introduces a very high order singularity at the solution, and it is doubtful whether this singularity can be overcome by using interpolation or other existing methods. Although there are numerous algorithms for solving Riccati equations, there still remains a need for algorithms which can operate efficiently on large problems and on parallel machines and which can be generalized easily to solve variants of Riccati equations. This thesis gives a new homotopy-based algorithm for solving Riccati equations on a shared memory parallel computer. The central part of the algorithm is the computation of the kernel of the Jacobian matrix, which is essential for the corrector iterations along the homotopy zero curve. Using a Schur decomposition the tensor product structure of various matrices can be efficiently exploited. The algorithm allows for efficient parallelization on shared memory machines. The linear-quadratic-Gaussian (LQG) theory has engendered a systematic approach to synthesize high performance controllers for nominal models of complex, multi-input multioutput systems and hence it is a breakthrough in modern control theory. Homotopy algorithms for both full and reduced-order LQG controller design problems with an H^∞ constraint on disturbance attenuation are developed. The H^∞ constraint is enforced by replacing the covariance Lyapunov equation by a Riccati equation whose solution gives an upper bound on H² performance. The numerical algorithm, based on homotopy theory, solves the necessary conditions for a minimum of the upper bound on H² performance. The algorithms are based on two minimal parameter formulations: Ly, Bryson, and Cannon's 2 X 2 block parametrization and the input normal Riccati form parametrization. An over-parametrization formulation is also proposed. Numerical experiments suggest that the combination of a globally convergent homotopy method with a minimal parameter formulation applied to the upper bound minimization gives excellent results for mixed-norm synthesis. / Ph. D.

Vertical Takeoff and Landing (VTOL) Unmanned Aerial Vehicles (UAVs) possess several beneficial attributes, including requiring minimal space to takeoff, hover, and land. The tailsitter is a type of VTOL airframe that combines the benefits of VTOL capability with the ability to achieve efficient horizontal flight. One type of tailsitter, the Quadrotor Biplane (QRBP), can transition the vehicle from hover as a quadrotor to horizontal flight as a biplane. The vehicle used in this thesis is a QRBP designed with special considerations for fully autonomous operation in an outdoor environment in the presence of model uncertainties. QRBPs undergo a rotation of 90° about its pitch axis during transition from vertical to horizontal flight that induces strong aerodynamic forces that are difficult to model, thus necessitating the use of a robust control method to overcome the resulting uncertainties in the model. A feedback-linearizing controller augmented with an H-Infinity robust control is developed to regulate the altitude and pitch angle of the vehicle for the whole flight regime, including the ascent, transition forward, and landing. The performance of the proposed control design is demonstrated through numerical simulations in MATLAB and outdoor flight tests. The H-Infinity controller successfully tracks the prescribed trajectory, demonstrating its value as a computationally inexpensive, robust control technique for QRBP tailsitter UAVs. / Master of Science / Vertical Takeoff and Landing (VTOL) Unmanned Aerial Vehicles (UAVs) are a special type of UAV that can takeoff, hover, and land vertically, which lends several benefits. VTOL aircraft have recently gained popularity due to their potential to serve as fast and efficient payload delivery vehicles for e-commerce. One type of VTOL aircraft, the Quadrotor Biplane (QRBP) combines the ability of a quadrotor aircraft to hover, with the efficient horizontal flight of a biplane. Such a vehicle is able to takeoff and land in confined spaces, and also travel large distances on a single battery. However, the takeoff maneuver of a QRBP involves pitching from vertical to horizontal flight, which causes the vehicle to experience strong aerodynamic effects that are difficult to accurately model. Thus, to autonomously perform this unique maneuver, a robust control technique is necessary. A robust UAV controller is one that functions even when there is a degree of uncertainty in the predicted behavior of the vehicle, such as differences between estimated and actual vehicle parameters, or the presence of external disturbances such as wind. Therefore, a robust controller known as H-Infinity is developed to regulate the altitude and pitch angle of the QRBP as it takes off, transitions to forward flight, flies as a biplane, transitions back to vertical flight, and lands. The performance of the proposed control design is validated using numerical simulations performed in MATLAB, and flight tests. The H-Infinity controller successfully tracks the prescribed trajectory, demonstrating its value as a reliable, computationally inexpensive, robust control technique for QRBP UAVs.

Emmanuel Levinas élabore une éthique en rupture avec la logique, la phénoménologie et l’ontologie. L’idée de l’infini, qu’il emprunte aux Méditations métaphysiques de Descartes, accomplit cette rupture et produit une philosophie d’un genre nouveau. Notre travail entend ainsi déterminer la façon dont cette idée permet à l’éthique de penser au sein de la philosophie une intrigue qui l’excède. L’idée de l’infini fait l’objet d’un triple discours : elle exprime la forme d’une pensée pensant plus qu’elle ne pense, elle décrit le sens de ce paradoxe dans la responsabilité infinie pour le visage de l’autre homme, et elle pratique une emphase conduisant au-delà de l’être. L’entrelacement de ces trois discours dans l’éthique de l’idée d’infini introduit en philosophie une question plus originaire que la question de l’être : la mise en question de mon être par l’infinie parole d’autrui, la mesure de mon être à l’aune de la question de l’infini, qui me vient à l’idée dans une exigence de justice. / The ethics that Emmanuel Levinas develops breaks with logic, phenomenology and ontology. The idea of infinity borrowed from Descartes’s Meditations on first Philosophy accomplishes this rupture and produces a new kind of philosophy. Our thesis aims to define the role played by the idea of infinity in enabling ethics to think within philosophy an intrigue which exceeds philosophy. This idea is subject to a triple discourse: it expresses the structure of a thought thinking more than it can think, it describes the meaning of this paradox as an infinite responsibility towards the other, and produces an emphasis that goes beyond essence. The unification of these three discourses within the ethics of the idea of infinity introduces a philosophical question that is more radical than the question of being: the question of infinity, which puts my being into question. Infinity comes to mind as a call for justice.

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I examine the way in which, in his work of the 1840s, Ruskin uses methods and assumptions derived from eighteenth-century Materialist, Mechanist and Vitalist Natural Philosophy, especially his assertion that the meanings which he reads into natural phenomena are objectively present and can be quantified, and the way in which therefore aesthetic concepts, responses and judgements can be quantified, and their values fixed. I examine the ways in which Ruskin seeks to demonstrate the relationship between the unity of Nature and the Multipilicity of Phenomena, not only as existing objectively in the external world, but also as reflected in the paintings of Turner. I suggest that his attempt at demonstration features a problematic relationship between his accounting for a material reality and the spiritual significances which he sees as immanent in it, and that resistance to the dynamism of contemporary industrial and social change is implicit in his celebration of an eternalised natural order. I examine four features of his correspondence during the 1840s: his dealings in the art market, his outright opposition to a number of modern developments, his urgent desires to see his favourite European architectural heritage preserved, and his strident xenophobia, and suggest relationships between the last two and his resistance to the modern. I examine the shift in his interests in the 1840s and 1850s from Nature and Art to Architecture and Man, and thence to Political Economy, and examine available accounts which rely too heavily on references to his psychological development, or on his claims to regular epiphanies, or on a significant shift in focus which can be explained by revealing the internal continuities in his work. I conclude with an attempt to demonstrate that what I have called the "broad sweep" approach obscures the confusions and contradictions in his position in the late 1840s and 1850s, and suggest that his social and intellectual inheritance, which is of a highly conservative and unremittingly paternalistic nature, crucially limits his work as ~ social critic. I offer three appendices: on the problem of the relationship between the Unity of Nature and the Multiplicity of Phenomena as that had been addressed in the Natural Philosophy on whose assumptions Ruskin draws; on eighteenth century Materialist, Mechanist and Vitalist theories of matter; and on the work of Edmund Burke and Sir Charles Bell.