Since Frege's predicate logical transcription of Aristotelian categorical logic, the standard sem... more Since Frege's predicate logical transcription of Aristotelian categorical logic, the standard semantics of Aristotelian logic considers terms as standing for sets of individuals. From a philosophical standpoint, this extensional model poses problems: There exist serious doubts that Aristotle's terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this question-they looked at terms as standing for concepts ("Begriffe"). In 1972, Corcoran presented a formal system for Aristotelian logic containing a calculus of natural deduction, while, with respect to semantics, he still made use of an extensional interpretation. In this paper we deal with a simple intensional semantics for Corcoran's syntax-intensional in the sense that no individuals are needed for the construction of a complete Tarski model of Aristotelian syntax. Instead, we view concepts as containing or excluding other, "higher" concepts-corresponding to the idea which Leibniz used in the construction of his characteristic numbers. Thus, this paper is an addendum to Corcoran's work, furnishing his formal syntax with an adequate semantics which is free from presuppositions which have entered into modern interpretations of Aristotle's theory via predicate logic. §1. Introduction. 1.1. Historical remarks. In 1972 Corcoran presented a formal system for Aristotelian logic containing a calculus of natural deduction (Corcoran, 1972b). The basic building blocks of this system are terms, as in Aristotle's works, as well as in those of his successors up to but not including Frege. 1 Corcoran's whole system does not contain variables or constants for individuals, and this conforms to the mainstream of the tradition in Aristotelian logic from Aristotle himself via Leibniz and Kant up to the first "modern" term logical system of Łukasiewicz (1957).
2016 Fourth International Conference on 3D Vision (3DV), 2016
Classical formulations of the shape matching problem involve the definition of a matching cost th... more Classical formulations of the shape matching problem involve the definition of a matching cost that directly depends on the action of the desired map when applied to some input data. Such formulations are typically one-sided-they seek for a mapping from one shape to the other, but not vice versa. In this paper we consider an unbiased formulation of this problem, in which we solve simultaneously for a low-distortion map relating the two given shapes and its inverse. We phrase the problem in the spectral domain using the language of functional maps, resulting in an especially compact and efficient optimization problem. The benefits of our proposed regularization are especially evident in the scarce data setting, where we demonstrate highly competitive results with respect to the state of the art.
Numerische Methoden bei Optimierungsaufgaben, 1973
Es gibt eine Reihe von Moglichkeiten, Kontrollprobleme als Optimierungsaufgaben in geeigneten Fun... more Es gibt eine Reihe von Moglichkeiten, Kontrollprobleme als Optimierungsaufgaben in geeigneten Funktionenraumen zu formulieren, s. z.B. NEUSTADT [6]. Der hier gewahlte Ansatz gestattet es, Kontrollaufgaben sowohl theoretisch (Existenz optimaler Steuerungen) zu behandeln, als auch numerische Verfahren zur Berechnung dieser Steuerungen zu untersuchen.
Numerische Methoden bei Optimierungsaufgaben, 1974
E und F seien reelle Hilbertraume. Mit L(E) bzw. L(E, F) bezeichnen wir wie ublich die Raume alle... more E und F seien reelle Hilbertraume. Mit L(E) bzw. L(E, F) bezeichnen wir wie ublich die Raume aller linearen und beschrankten Operatoren, die auf E erklart sind und E in sich bzw. in F abbilden.
In 1679, Leibniz wrote nine manuscripts on three different arithmetical models of Aristotelian lo... more In 1679, Leibniz wrote nine manuscripts on three different arithmetical models of Aristotelian logic. This was a part of his project of a "calculus universalis". First we show the precise relations of these three models to each other by presenting three criteria which serve the purpose of classifiying models of Aristotelian logic. This facilitates the understanding of Leibniz' constructions. Our method is of special value for the sophisticated third model, the domain of which consists of pairs of natural numbers. We present a simple approach to Leibniz' definitions which on first sight appear complicated. We show that it is possible to deduce, from the "universal positive" relation a (= "All ..."), the other three "Aristotelian relations" i, o, and e.-It has always been difficult to understand the exact nature of Leibniz' characteristic numbers because of his misleading nomenclature, since he utilized the signs + and-in order to designate the "positive" and "negative" part of a characteristic number pair. We present a new interpretation of Leibniz' symbolism, showing that the number pairs should be interpreted as numerator and denominator of a rational number. Thus we can identify the last model as the natural extension of the first and second one, showing the continuity in Leibniz' different attempts towards an arithmetization of logic.-We close our paper by discussing two well known problematic aspects of Leibniz' characteristic numbers, formulating two open questions concerning the formal structure of the system. ZUSAMMENFASSUNG: Im Rahmen seines Projektes eines "calculus universalis" entwarf Leibniz Anfang 1679 in neun Texten drei unterschiedliche Modelle der aristotelischen Logik mit Hilfe von Zahlen. Durch diese von ihm erfundenen "charakteristischen Zahlen" wollte Leibniz die Schlussweisen der aristotelischen Logik auf rein arithmetische Rechnungen reduzieren. In der vorliegenden Arbeit zeigen wir genau, wie die drei Modelle untereinander zusammenhängen bzw. aufeinander aufbauen. Zu diesem Zweck geben wir drei Kriterien an, mit Hilfe derer sich Modelle der aristotelischen Logik klassifizieren lassen. Zum einen können wir dadurch die Leibnizschen Definitionen leicht nachvollziehen sowie auch die Stärken und Schwächen der einzelnen Modelle präzise beschreiben. Besonders für das ausgefeilte letzte Modell, dessen Grundbereich aus Paaren natürlicher Zahlen besteht, liefert unsere Methode einen ganz natürlichen Zugang zu den Leibnizschen Definitionen, die im Original recht sperrig wirken. Wir zeigen insbesondere-was Leibniz nicht herausgestellt hat-, dass allein aus der universell-positiven alle anderen drei "aristotelischen Relationen" herleitbar sind. Ein Grund für manche Schwierigkeiten mit dem Verständnis der Leibnizschen charakteristischen Zahlen ist die von ihm verwendete Nomenklatur. Der problematischen Schreibweise mit Vorzeichen + und-, die in der Vergangenheit manchen Kommentator auf die falsche Fährte gelockt hat, geben wir eine neue Deutung: Dadurch entpuppt sich das letzte Modell als ganz natürliche Erweiterung des vorangehenden vom Grundbereich der natürlichen in den Bereich der positiven rationalen Zahlen.-Mit der vorliegenden Arbeit hoffen wir, der Verwendung charakteristischer Zahlen als theoretisches Instrument für die Untersuchung der aristotelischen Logik einen neuen Impuls geben zu können. Deshalb diskutieren wir am Schluss der Arbeit noch die aus der Literatur bekannten Stärken und Schwächen des Leibnizschen Modells und formulieren zwei offene Fragen zu diesem Komplex. 7 A, N. 56, S. 182. "It is necessary for the universal positive proposition to be true, that the subject number can be divided exactly without remainder by the predicate number". 8 A, N. 57, p. 199-200. 9 "... ita ut generis notio sit pars, speciei notio sit totum, componitur enim ex genere et differentia." (A, N. 57, 11, p. 199). 10 N 56, 17, p. 182. 11 "If, for example, we assume that the item ‚animal' is expressed by means of the number 2 (or, in general, by a), and the item ‚rational' by means of the number 3 (or, in general, by r), then ‚man' is expressed by 2 ⋅3, i.e. 6, as the result of the product of 2 by 3 (or, in general, by the number a⋅r)". 12 Here we do not include zero in the set of natural numbers.
This chapter will be devoted to the study of the problem pairs (P)-(D) and (PA)-(DA) in a special... more This chapter will be devoted to the study of the problem pairs (P)-(D) and (PA)-(DA) in a special but important case, namely when the moment generating functions al, ••. ,a n form a so-called Chebyshev system. k • f .
A major topic of this chapter is the derivation of “strong” duality results, i.e. theorems which ... more A major topic of this chapter is the derivation of “strong” duality results, i.e. theorems which specify when v(D) = v(P). Another important topic is the existence of solutions to the problems (P) and (D). We shall give two strong duality theorems, namely (9) of §10 and (7) of §11. They can be used to verify strong duality in most linear optimization problems occurring in practice.
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2015
We construct an extension of spectral and diffusion geometry to multiple modalities through simul... more We construct an extension of spectral and diffusion geometry to multiple modalities through simultaneous diagonalization of Laplacian matrices. This naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real examples of manifold learning, object classification, and clustering, showing that the joint spectral geometry better captures the inherent structure of multi-modal data. We also show the relation of many previous approaches for multimodal manifold analysis to our framework.
In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting o... more In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting operators (CCO) to a given pair of Laplacians. The CCOs are jointly diagonalizable and hence have the same eigenbasis. Our construction naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real examples of applications in dimensionality reduction, shape analysis, and clustering, demonstrating that our method better captures the inherent structure of multi-modal data.
The temperature distribution of a given body $\Omega $ in $\mathbb{R}^N $, $N \geqq 1$, is contro... more The temperature distribution of a given body $\Omega $ in $\mathbb{R}^N $, $N \geqq 1$, is controlled via the temperature ${\bf u}$ of the medium surrounding $\Omega $. The task is to choose ${\bf u}$ (subject to certain restrictions) in such a way that the temperature distribution at time $t = T$ of $\Omega $ comes as close as possible (with respect to the supremum-norm, i.e., the norm in $C(\bar \Omega )$) to a fixed given function. We study the question of existence of optimal controls and prove the bang-bang principle for two different control problems (in the first case, the control ${\bf u}$ is time- and space-dependent and in the other one ${\bf u}$ is only time-dependent). In the last part of this paper, we formulate an abstract minimum-norm problem and derive general theorems on the existence and characterization of optimal controls.
Since Frege's predicate logical transcription of Aristotelian categorical logic, the standard sem... more Since Frege's predicate logical transcription of Aristotelian categorical logic, the standard semantics of Aristotelian logic considers terms as standing for sets of individuals. From a philosophical standpoint, this extensional model poses problems: There exist serious doubts that Aristotle's terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this question-they looked at terms as standing for concepts ("Begriffe"). In 1972, Corcoran presented a formal system for Aristotelian logic containing a calculus of natural deduction, while, with respect to semantics, he still made use of an extensional interpretation. In this paper we deal with a simple intensional semantics for Corcoran's syntax-intensional in the sense that no individuals are needed for the construction of a complete Tarski model of Aristotelian syntax. Instead, we view concepts as containing or excluding other, "higher" concepts-corresponding to the idea which Leibniz used in the construction of his characteristic numbers. Thus, this paper is an addendum to Corcoran's work, furnishing his formal syntax with an adequate semantics which is free from presuppositions which have entered into modern interpretations of Aristotle's theory via predicate logic. §1. Introduction. 1.1. Historical remarks. In 1972 Corcoran presented a formal system for Aristotelian logic containing a calculus of natural deduction (Corcoran, 1972b). The basic building blocks of this system are terms, as in Aristotle's works, as well as in those of his successors up to but not including Frege. 1 Corcoran's whole system does not contain variables or constants for individuals, and this conforms to the mainstream of the tradition in Aristotelian logic from Aristotle himself via Leibniz and Kant up to the first "modern" term logical system of Łukasiewicz (1957).
2016 Fourth International Conference on 3D Vision (3DV), 2016
Classical formulations of the shape matching problem involve the definition of a matching cost th... more Classical formulations of the shape matching problem involve the definition of a matching cost that directly depends on the action of the desired map when applied to some input data. Such formulations are typically one-sided-they seek for a mapping from one shape to the other, but not vice versa. In this paper we consider an unbiased formulation of this problem, in which we solve simultaneously for a low-distortion map relating the two given shapes and its inverse. We phrase the problem in the spectral domain using the language of functional maps, resulting in an especially compact and efficient optimization problem. The benefits of our proposed regularization are especially evident in the scarce data setting, where we demonstrate highly competitive results with respect to the state of the art.
Numerische Methoden bei Optimierungsaufgaben, 1973
Es gibt eine Reihe von Moglichkeiten, Kontrollprobleme als Optimierungsaufgaben in geeigneten Fun... more Es gibt eine Reihe von Moglichkeiten, Kontrollprobleme als Optimierungsaufgaben in geeigneten Funktionenraumen zu formulieren, s. z.B. NEUSTADT [6]. Der hier gewahlte Ansatz gestattet es, Kontrollaufgaben sowohl theoretisch (Existenz optimaler Steuerungen) zu behandeln, als auch numerische Verfahren zur Berechnung dieser Steuerungen zu untersuchen.
Numerische Methoden bei Optimierungsaufgaben, 1974
E und F seien reelle Hilbertraume. Mit L(E) bzw. L(E, F) bezeichnen wir wie ublich die Raume alle... more E und F seien reelle Hilbertraume. Mit L(E) bzw. L(E, F) bezeichnen wir wie ublich die Raume aller linearen und beschrankten Operatoren, die auf E erklart sind und E in sich bzw. in F abbilden.
In 1679, Leibniz wrote nine manuscripts on three different arithmetical models of Aristotelian lo... more In 1679, Leibniz wrote nine manuscripts on three different arithmetical models of Aristotelian logic. This was a part of his project of a "calculus universalis". First we show the precise relations of these three models to each other by presenting three criteria which serve the purpose of classifiying models of Aristotelian logic. This facilitates the understanding of Leibniz' constructions. Our method is of special value for the sophisticated third model, the domain of which consists of pairs of natural numbers. We present a simple approach to Leibniz' definitions which on first sight appear complicated. We show that it is possible to deduce, from the "universal positive" relation a (= "All ..."), the other three "Aristotelian relations" i, o, and e.-It has always been difficult to understand the exact nature of Leibniz' characteristic numbers because of his misleading nomenclature, since he utilized the signs + and-in order to designate the "positive" and "negative" part of a characteristic number pair. We present a new interpretation of Leibniz' symbolism, showing that the number pairs should be interpreted as numerator and denominator of a rational number. Thus we can identify the last model as the natural extension of the first and second one, showing the continuity in Leibniz' different attempts towards an arithmetization of logic.-We close our paper by discussing two well known problematic aspects of Leibniz' characteristic numbers, formulating two open questions concerning the formal structure of the system. ZUSAMMENFASSUNG: Im Rahmen seines Projektes eines "calculus universalis" entwarf Leibniz Anfang 1679 in neun Texten drei unterschiedliche Modelle der aristotelischen Logik mit Hilfe von Zahlen. Durch diese von ihm erfundenen "charakteristischen Zahlen" wollte Leibniz die Schlussweisen der aristotelischen Logik auf rein arithmetische Rechnungen reduzieren. In der vorliegenden Arbeit zeigen wir genau, wie die drei Modelle untereinander zusammenhängen bzw. aufeinander aufbauen. Zu diesem Zweck geben wir drei Kriterien an, mit Hilfe derer sich Modelle der aristotelischen Logik klassifizieren lassen. Zum einen können wir dadurch die Leibnizschen Definitionen leicht nachvollziehen sowie auch die Stärken und Schwächen der einzelnen Modelle präzise beschreiben. Besonders für das ausgefeilte letzte Modell, dessen Grundbereich aus Paaren natürlicher Zahlen besteht, liefert unsere Methode einen ganz natürlichen Zugang zu den Leibnizschen Definitionen, die im Original recht sperrig wirken. Wir zeigen insbesondere-was Leibniz nicht herausgestellt hat-, dass allein aus der universell-positiven alle anderen drei "aristotelischen Relationen" herleitbar sind. Ein Grund für manche Schwierigkeiten mit dem Verständnis der Leibnizschen charakteristischen Zahlen ist die von ihm verwendete Nomenklatur. Der problematischen Schreibweise mit Vorzeichen + und-, die in der Vergangenheit manchen Kommentator auf die falsche Fährte gelockt hat, geben wir eine neue Deutung: Dadurch entpuppt sich das letzte Modell als ganz natürliche Erweiterung des vorangehenden vom Grundbereich der natürlichen in den Bereich der positiven rationalen Zahlen.-Mit der vorliegenden Arbeit hoffen wir, der Verwendung charakteristischer Zahlen als theoretisches Instrument für die Untersuchung der aristotelischen Logik einen neuen Impuls geben zu können. Deshalb diskutieren wir am Schluss der Arbeit noch die aus der Literatur bekannten Stärken und Schwächen des Leibnizschen Modells und formulieren zwei offene Fragen zu diesem Komplex. 7 A, N. 56, S. 182. "It is necessary for the universal positive proposition to be true, that the subject number can be divided exactly without remainder by the predicate number". 8 A, N. 57, p. 199-200. 9 "... ita ut generis notio sit pars, speciei notio sit totum, componitur enim ex genere et differentia." (A, N. 57, 11, p. 199). 10 N 56, 17, p. 182. 11 "If, for example, we assume that the item ‚animal' is expressed by means of the number 2 (or, in general, by a), and the item ‚rational' by means of the number 3 (or, in general, by r), then ‚man' is expressed by 2 ⋅3, i.e. 6, as the result of the product of 2 by 3 (or, in general, by the number a⋅r)". 12 Here we do not include zero in the set of natural numbers.
This chapter will be devoted to the study of the problem pairs (P)-(D) and (PA)-(DA) in a special... more This chapter will be devoted to the study of the problem pairs (P)-(D) and (PA)-(DA) in a special but important case, namely when the moment generating functions al, ••. ,a n form a so-called Chebyshev system. k • f .
A major topic of this chapter is the derivation of “strong” duality results, i.e. theorems which ... more A major topic of this chapter is the derivation of “strong” duality results, i.e. theorems which specify when v(D) = v(P). Another important topic is the existence of solutions to the problems (P) and (D). We shall give two strong duality theorems, namely (9) of §10 and (7) of §11. They can be used to verify strong duality in most linear optimization problems occurring in practice.
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2015
We construct an extension of spectral and diffusion geometry to multiple modalities through simul... more We construct an extension of spectral and diffusion geometry to multiple modalities through simultaneous diagonalization of Laplacian matrices. This naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real examples of manifold learning, object classification, and clustering, showing that the joint spectral geometry better captures the inherent structure of multi-modal data. We also show the relation of many previous approaches for multimodal manifold analysis to our framework.
In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting o... more In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting operators (CCO) to a given pair of Laplacians. The CCOs are jointly diagonalizable and hence have the same eigenbasis. Our construction naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real examples of applications in dimensionality reduction, shape analysis, and clustering, demonstrating that our method better captures the inherent structure of multi-modal data.
The temperature distribution of a given body $\Omega $ in $\mathbb{R}^N $, $N \geqq 1$, is contro... more The temperature distribution of a given body $\Omega $ in $\mathbb{R}^N $, $N \geqq 1$, is controlled via the temperature ${\bf u}$ of the medium surrounding $\Omega $. The task is to choose ${\bf u}$ (subject to certain restrictions) in such a way that the temperature distribution at time $t = T$ of $\Omega $ comes as close as possible (with respect to the supremum-norm, i.e., the norm in $C(\bar \Omega )$) to a fixed given function. We study the question of existence of optimal controls and prove the bang-bang principle for two different control problems (in the first case, the control ${\bf u}$ is time- and space-dependent and in the other one ${\bf u}$ is only time-dependent). In the last part of this paper, we formulate an abstract minimum-norm problem and derive general theorems on the existence and characterization of optimal controls.
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Papers by Klaus Glashoff