A. Kock - Academia.edu (original) (raw)

Papers by A. Kock

Central European Journal of Mathematics, 2006

In the context of Synthetic Differential Geometry, we discuss vector fields/ordinary differential... more In the context of Synthetic Differential Geometry, we discuss vector fields/ordinary differential equations as actions; in particular, we exploit function space formation (exponential spaces) in the category of actions.

Bulletin of the Australian Mathematical Society, 1982

Communications in Algebra, 1984

Theory and Applications of Categories, 1998

This paper grew out of a question of Lawvere, who asked whether categories of ordinary (1st or 2n... more This paper grew out of a question of Lawvere, who asked whether categories of ordinary (1st or 2nd) order differential equations may be seen as toposes. He also gave some indications how “fractional exponents” may be used to answer this question,[25].

MATHEMATICA SCANDINAVICA, 1967

Topology and its Applications, 1996

We elaborate a suggestion of Grothendieck, and study the invariant sheaves for a local equivalenc... more We elaborate a suggestion of Grothendieck, and study the invariant sheaves for a local equivalence relation on a space (e.g., a foliation). One of our purposes is to compare this to the standard model for the leaf-(quotient-)space of a foliation, given by the holonomy groupoid. To this end, we prove that, under suitable connectedness assumptions, Grothendieck's invariant sheaves can be described in terms of a closely related, but different, "monodromy" groupoid. Our second purpose is to prove that every Ctale groupoid arises this way.

Publisher Summary This chapter presents category-theoretic methods in logic; the focus is on mode... more Publisher Summary This chapter presents category-theoretic methods in logic; the focus is on model theory and set theory. It is organized by increasing richness of the doctrines involved. These doctrines are categorical analogs of fragments of logical theories that have sufficient category-theoretic structure for their models to be described as functors. The introduction of the categorical notion of algebraic theory led to a systematic theory of relative interpretations of one equational theory into another, as well as a theory about the categories (or varieties) of algebras for these, and their relationship. This progress springs from having a presentation-invariant notion of equational (or algebraic) theory. The chapter also deals with doctrines of equational, Cartesian, finitary coherent, and infinitary coherent logic. Higher order logic and set theory are also discussed in the chapter.

Cah. Topol. Géom. Différ, 1984

Strong infinitesimal linearity, with applications to strong difference and affine connections Cah... more Strong infinitesimal linearity, with applications to strong difference and affine connections Cahiers de topologie et géométrie différentielle catégoriques, tome 25, n o 3 (1984), p. 311-324 <http://www.numdam.org/item?id=CTGDC_1984__25_3_311_0> © Andrée C. Ehresmann et les auteurs, 1984, tous droits réservés. L'accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Communications in Algebra, 1975

... Page 2. 546 KOCK ... rank, ... ) are formulated in that fragment of 1st order lan-guage which... more ... Page 2. 546 KOCK ... rank, ... ) are formulated in that fragment of 1st order lan-guage which Joyal,Reyes and others have called geometric logic, [121.) I n $1, we give the preliminaries about tensor products, duality, and exterior powers, in a very general setting; §2 ...

Arxiv preprint math/0407242, 2004

Abstract: The aim of this paper is to develop the theory of distributions, not necessarily of com... more Abstract: The aim of this paper is to develop the theory of distributions, not necessarily of compact support, in a topos model of Synthetic Differential Geometry, the so-called" Cahiers Topos". As an application, we study the evolution through time of a heat distribution. In the ...

Journal of Pure and Applied Algebra, 1992

Kock, A. and I. Moerdijk, Every &endue comes from a local equivalence relation, Journal of Pure a... more Kock, A. and I. Moerdijk, Every &endue comes from a local equivalence relation, Journal of Pure and Applied Algebra 82 (1992) 155-174. We first prove that, under suitable connectedness assumptions, the equivariant sheaves for a local equivalence relation on a space (or a locale) form an &endue topos. Our main result is that conversely, every &endue can be obtained in this way.

The simplest notion by which a theory of function spaces may be formulated is that of cartesian c... more The simplest notion by which a theory of function spaces may be formulated is that of cartesian closed categories. To realize this concretely for spaces of smooth (= C) functions, several notions of diffeological spaces and convenient vector spaces have been developed, besides the whole body of topos theory. Topos theory in particular provides for toposes containing the category of smooth manifolds as full subcategory. In fact, Grothendieck’s “Smooth Topos” is closely related to the category of diffeological spaces. The special features of Convenient Vector Spaces were utilized by the present authors back in the 1980’s for a more elaborate topos , cf. [12], [14]. The topos there (Dubuc’s “Cahiers Topos”) in fact accomodates synthetic differential geometry. The present work is a continuation of our work from the 80’s, and is motivated by the desire to have a synthetic theory of some of the fundamental partial differential equations, like the heat equation. This forced us to sort out ...

Bulletin of the Australian Mathematical Society, 1986

In the context of synthetic differential geometry, we prove that group valued 1-forms on the unit... more In the context of synthetic differential geometry, we prove that group valued 1-forms on the unit interval are exact, provided the group in question is a Lie group. This exactness is the basic assumption in a previous paper by the author on differential forms with values in groups.

Semiconductor Science and Technology

... K, Bethea CG, Walker 1 and Malik R J 1987 Appl. Phys. Lerr. S1 934 Lyon S 1990 Sur5 Sci. 228 ... more ... K, Bethea CG, Walker 1 and Malik R J 1987 Appl. Phys. Lerr. S1 934 Lyon S 1990 Sur5 Sci. 228 508 Berthold K, Beinstingl W, Berger R and Gornik E Appl. Phys. Lett. 48 526 Berthold K, Beinstingl W and Gornik E 1987 Opt. 17 la 1986 Lett. __ Y , [SI Jest1 M. Kock A, Beinstingl ...

CLEO/Europe Conference on Lasers and Electro-Optics, 1998

Lasers and Electro-Optics Society, IEEE LEOS Annual Meeting, 1995

The potentials of a multi quantum-well (MQW)-modulator-diode for frequency tuning of VCSELs based... more The potentials of a multi quantum-well (MQW)-modulator-diode for frequency tuning of VCSELs based on the quantum-confined Stark-effect (QCSE) as well as the impact of frequency tuning on output power and transversal mode spectra are discussed. This is of interest for high performance frequency tuneable VCSELs, where a continuous modulation of the laser wavelength is required with constant laser power and

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Conference on Lasers and Electro-Optics Europe, 1994

Le Journal De Physique Colloques, 1988

We have investigated the light emission from forward and reverse biased sinusoidally structured A... more We have investigated the light emission from forward and reverse biased sinusoidally structured Ag/n-GaAs Schottky diodes. These Schottky junctions provide increased light emission due to the radiative decay of excited surface plasmon polaritons, resulting in drastically enhanced quantum efficiency. Conversely surface plasma modes excited at the surface of a semiconductor-metal-insulator structure provide selective coupling of light into the semiconductor with increased quantum efficiency. It is shown, that Schottky diodes can be used as photodetectors selective to polarization, frequency and angle of incident light.

Applied Physics Letters, 1992

... the wells. Most experiments to date have been performed by shining light on the edge of the s... more ... the wells. Most experiments to date have been performed by shining light on the edge of the sample. However, for most technical applications planar device ge-ometry and large area ilhunination are essential. Therefore we ...

Lecture Notes in Mathematics, 1991

Central European Journal of Mathematics, 2006

In the context of Synthetic Differential Geometry, we discuss vector fields/ordinary differential... more In the context of Synthetic Differential Geometry, we discuss vector fields/ordinary differential equations as actions; in particular, we exploit function space formation (exponential spaces) in the category of actions.

Bulletin of the Australian Mathematical Society, 1982

Communications in Algebra, 1984

Theory and Applications of Categories, 1998

This paper grew out of a question of Lawvere, who asked whether categories of ordinary (1st or 2n... more This paper grew out of a question of Lawvere, who asked whether categories of ordinary (1st or 2nd) order differential equations may be seen as toposes. He also gave some indications how “fractional exponents” may be used to answer this question,[25].

MATHEMATICA SCANDINAVICA, 1967

Topology and its Applications, 1996

We elaborate a suggestion of Grothendieck, and study the invariant sheaves for a local equivalenc... more We elaborate a suggestion of Grothendieck, and study the invariant sheaves for a local equivalence relation on a space (e.g., a foliation). One of our purposes is to compare this to the standard model for the leaf-(quotient-)space of a foliation, given by the holonomy groupoid. To this end, we prove that, under suitable connectedness assumptions, Grothendieck's invariant sheaves can be described in terms of a closely related, but different, "monodromy" groupoid. Our second purpose is to prove that every Ctale groupoid arises this way.

Publisher Summary This chapter presents category-theoretic methods in logic; the focus is on mode... more Publisher Summary This chapter presents category-theoretic methods in logic; the focus is on model theory and set theory. It is organized by increasing richness of the doctrines involved. These doctrines are categorical analogs of fragments of logical theories that have sufficient category-theoretic structure for their models to be described as functors. The introduction of the categorical notion of algebraic theory led to a systematic theory of relative interpretations of one equational theory into another, as well as a theory about the categories (or varieties) of algebras for these, and their relationship. This progress springs from having a presentation-invariant notion of equational (or algebraic) theory. The chapter also deals with doctrines of equational, Cartesian, finitary coherent, and infinitary coherent logic. Higher order logic and set theory are also discussed in the chapter.

Cah. Topol. Géom. Différ, 1984

Strong infinitesimal linearity, with applications to strong difference and affine connections Cah... more Strong infinitesimal linearity, with applications to strong difference and affine connections Cahiers de topologie et géométrie différentielle catégoriques, tome 25, n o 3 (1984), p. 311-324 <http://www.numdam.org/item?id=CTGDC_1984__25_3_311_0> © Andrée C. Ehresmann et les auteurs, 1984, tous droits réservés. L'accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Communications in Algebra, 1975

... Page 2. 546 KOCK ... rank, ... ) are formulated in that fragment of 1st order lan-guage which... more ... Page 2. 546 KOCK ... rank, ... ) are formulated in that fragment of 1st order lan-guage which Joyal,Reyes and others have called geometric logic, [121.) I n $1, we give the preliminaries about tensor products, duality, and exterior powers, in a very general setting; §2 ...

Arxiv preprint math/0407242, 2004

Abstract: The aim of this paper is to develop the theory of distributions, not necessarily of com... more Abstract: The aim of this paper is to develop the theory of distributions, not necessarily of compact support, in a topos model of Synthetic Differential Geometry, the so-called" Cahiers Topos". As an application, we study the evolution through time of a heat distribution. In the ...

Journal of Pure and Applied Algebra, 1992

Kock, A. and I. Moerdijk, Every &endue comes from a local equivalence relation, Journal of Pure a... more Kock, A. and I. Moerdijk, Every &endue comes from a local equivalence relation, Journal of Pure and Applied Algebra 82 (1992) 155-174. We first prove that, under suitable connectedness assumptions, the equivariant sheaves for a local equivalence relation on a space (or a locale) form an &endue topos. Our main result is that conversely, every &endue can be obtained in this way.

The simplest notion by which a theory of function spaces may be formulated is that of cartesian c... more The simplest notion by which a theory of function spaces may be formulated is that of cartesian closed categories. To realize this concretely for spaces of smooth (= C) functions, several notions of diffeological spaces and convenient vector spaces have been developed, besides the whole body of topos theory. Topos theory in particular provides for toposes containing the category of smooth manifolds as full subcategory. In fact, Grothendieck’s “Smooth Topos” is closely related to the category of diffeological spaces. The special features of Convenient Vector Spaces were utilized by the present authors back in the 1980’s for a more elaborate topos , cf. [12], [14]. The topos there (Dubuc’s “Cahiers Topos”) in fact accomodates synthetic differential geometry. The present work is a continuation of our work from the 80’s, and is motivated by the desire to have a synthetic theory of some of the fundamental partial differential equations, like the heat equation. This forced us to sort out ...

Bulletin of the Australian Mathematical Society, 1986

In the context of synthetic differential geometry, we prove that group valued 1-forms on the unit... more In the context of synthetic differential geometry, we prove that group valued 1-forms on the unit interval are exact, provided the group in question is a Lie group. This exactness is the basic assumption in a previous paper by the author on differential forms with values in groups.

Semiconductor Science and Technology

... K, Bethea CG, Walker 1 and Malik R J 1987 Appl. Phys. Lerr. S1 934 Lyon S 1990 Sur5 Sci. 228 ... more ... K, Bethea CG, Walker 1 and Malik R J 1987 Appl. Phys. Lerr. S1 934 Lyon S 1990 Sur5 Sci. 228 508 Berthold K, Beinstingl W, Berger R and Gornik E Appl. Phys. Lett. 48 526 Berthold K, Beinstingl W and Gornik E 1987 Opt. 17 la 1986 Lett. __ Y , [SI Jest1 M. Kock A, Beinstingl ...

CLEO/Europe Conference on Lasers and Electro-Optics, 1998

Lasers and Electro-Optics Society, IEEE LEOS Annual Meeting, 1995

The potentials of a multi quantum-well (MQW)-modulator-diode for frequency tuning of VCSELs based... more The potentials of a multi quantum-well (MQW)-modulator-diode for frequency tuning of VCSELs based on the quantum-confined Stark-effect (QCSE) as well as the impact of frequency tuning on output power and transversal mode spectra are discussed. This is of interest for high performance frequency tuneable VCSELs, where a continuous modulation of the laser wavelength is required with constant laser power and

[

Conference on Lasers and Electro-Optics Europe, 1994

Le Journal De Physique Colloques, 1988

We have investigated the light emission from forward and reverse biased sinusoidally structured A... more We have investigated the light emission from forward and reverse biased sinusoidally structured Ag/n-GaAs Schottky diodes. These Schottky junctions provide increased light emission due to the radiative decay of excited surface plasmon polaritons, resulting in drastically enhanced quantum efficiency. Conversely surface plasma modes excited at the surface of a semiconductor-metal-insulator structure provide selective coupling of light into the semiconductor with increased quantum efficiency. It is shown, that Schottky diodes can be used as photodetectors selective to polarization, frequency and angle of incident light.

Applied Physics Letters, 1992

... the wells. Most experiments to date have been performed by shining light on the edge of the s... more ... the wells. Most experiments to date have been performed by shining light on the edge of the sample. However, for most technical applications planar device ge-ometry and large area ilhunination are essential. Therefore we ...

Lecture Notes in Mathematics, 1991