Hans-Joachim Arnold

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Hans-Joachim Arnold

Hans-Joachim Felix Arnold (born March 31, 1932 in Berlin ; † February 20, 2006 in Mülheim an der Ruhr ) was a German mathematician and university professor . The focus of his work was the connection of universal algebra and geometry , he founded the geometric relation algebra .

Life

Arnold studied at the University of Hamburg from 1952 to 1958 and received his doctorate from Emanuel Sperner in 1965 with a thesis on distant spaces with affine spaces . In 1970 he completed his habilitation with Sperner with his thesis The Geometry of Rings in the Framework of General Affine Structures . In 1966 he became an assistant at the Ruhr University in Bochum , and in 1973 he was appointed as the founding senator for the subject of mathematics during the establishment of the then comprehensive university in Duisburg . Arnold founded the journal Results in Mathematics together with Heinrich Wefelscheid in 1977 .

plant

Arnold was able to finally solve a problem of the algebraization of unnecessary Desargue affine and projective geometries with the relational algebraic calculus of relatives and multigroups that he introduced . Conventional structures such as groupoids , quasi-modules or ternary bodies algebraize weakly affine geometries , i.e. in particular also non-Desarguean affine planes , and in turn can also generate these geometries. In all cases, however, the synonymity condition is violated because of missing coordinate areas or because of dependencies on the choice of a coordinate system required for the transition process. Only with the affine relatives, which consist of a set of relations that operate on the set of points of the geometry presented, do the transition processes of algebraization and geometrization become synonymous, i.e. H. except for isomorphism, around each other.

Another advantage of the relational algebraic way of speaking lies in its constructive expandability: The language of geometric relational algebra is suitable for specifying simple calculation rules that are equivalent to extensive additional geometric axioms (closing clauses). The two-stage (H2) homogeneity rule developed by Arnold is equivalent to the constructability of parallel-like triangles, i.e. to the Tamaschke axiom . His three-level (H3) -homogeneity rule finds its equivalent on the geometrical side in the validity of Desargues' large affine theorem in the plane. Through an antisymmetry of the operators in the affine directional relatives, Arnold was able to describe the affine geometries arranged in the sense of David Hilbert synonymously. His doctoral students Roland Soltysiak, Andreas Kopp and Chandrasekara Senevirathne then succeeded in the synonymous correspondence of almost body geometries, line geometries and the - in short: semi-ordered - affine geometries arranged in the sense of Emanuel Sperner through almost-affine relatives, line relatives and affine orientation relatives.

In all of these geometries, time does not yet play a role, but Arnold also succeeds in dynamizing the affine relatives by including time structures with the rule relatives . While complex mathematical methods of differential equation systems , differential geometry or differential algebra are used to analyze and model dynamic systems , he provides a new mathematical language for time-discrete and continuous systems with the "rule-relatives" synonymous with the general system term by Eduardo D. Sontag. With this approach, his doctoral students Peter Stemper, Marc Schleuter and Dirk Wetscheck were able to capture example classes of linear, nonlinear and fuzzy systems from control theory using the same mathematical method; Axel Sauerland showed the isomorphism of rule relatives, defined by state-homogeneous and input-homogeneous bilinear systems, to Desargue's affine relatives.

Arnold initially described projective geometries as synonymous with so-called (three-dimensional) projective multigroups. With the 2x2 relations operating on this point set, which in turn define a projective (2x2) -relative synonymously, the description of the large projective theorem of Desargues in the plane succeeds through the constructive expandability to a (H2x2) -homogeneity rule.

Arnold's affine or projective relatives in the conceptual world of algebra and affine or projective geometries turn out to be two different ways of speaking for one and the same state of affairs. In addition, with the relative, he also succeeds in providing a mathematical way of describing cognitive theory . Action theory concepts and cognitive aspects in the regulation of simple dynamic systems are also mathematized by him using relation theory methods.

Fonts

  • About distant spaces of weakly affine spaces. In: Treatises from the Mathematical Seminar of the University of Hamburg. Volume 30, Universität Berlin, Hamburg 1967, pp. 75-105, doi: 10.1007 / BF02993993 .
  • About a class of Sperner's quasi-modules. In: Treatises from the Mathematical Seminar of the University of Hamburg. Volume 31, Universität Berlin, Hamburg 1967, pp. 206-212, doi: 10.1007 / BF02992400 .
  • Algebraic and geometric characterization of the weakly affine vector spaces over nearly fields. In: Treatises from the Mathematical Seminar of the University of Hamburg. Volume 32, Universität Berlin, Hamburg 1968, pp. 73-88, doi: 10.1007 / BF02993915 .
  • Shell operations and transfinite Steinitzer exchange set. In: Treatises from the Mathematical Seminar of the University of Hamburg. Volume 33, Universität Berlin, Hamburg 1969, pp. 32-42, doi: 10.1007 / BF02992802 .
  • The geometry of the rings in the context of general affine structures. In: Hamburg individual mathematical writings. Vandenhoeck & Ruprecht, Göttingen New Series, Issue 4, 1971.
  • A way to the geometry of rings. In: Journal of Geometry. Volume 1, issue 2, 1971, pp. 155-167, doi: 10.1007 / BF02150269 .
  • Connection of geometric and algebraic structures. In: Treatises from the Mathematical Seminar of the University of Hamburg. Volume 37, Universität Berlin, Hamburg 1972, pp. 1-5, doi: 10.1007 / BF02993894 .
  • The projective closure of affine gemotria with the help of relation-theoretic methods. In: Treatises from the Mathematical Seminar of the University of Hamburg. Volume 40, Universität Berlin, Hamburg 1974, pp. 197-214, doi: 10.1007 / BF02993598 .
  • A relation-theoretic algebraization of arranged affine and projective geometries. In: Treatises from the Mathematical Seminar of the University of Hamburg. Volume 45, Universität Berlin, Hamburg 1976, pp. 3-60, doi: 10.1007 / BF02992902 .
  • Relational groupings within the framework of Piaget's developmental psychology. In: Contributions to Geometric Algebra. 1977, pp. 361-366, doi : 10.1007 / 978-3-0348-5573-0_49 .
  • For the algebraization of general affine and associated projective structures with the help of a vector calculus. In: Contributions to Geometric Algebra. 1977, pp. 25-29, doi : 10.1007 / 978-3-0348-5573-0_2 .
  • To characterize Sperner's spaces, which are homogeneous in two points. In: Journal of Geometry. Volume 9, issue 1-2, 1977, pp. 9-17, doi: 10.1007 / BF01918053 .
  • About the structure of Sperner's distributive spaces, which are homogeneous in two respects, with special consideration of their distant spaces. In: Archives of Mathematics. Volume 30, issue 1, 1978, pp. 551-560, doi: 10.1007 / BF01226100 .
  • Directional algebras. In: Contributions to Geometry. 1979, pp. 379-382, doi : 10.1007 / 978-3-0348-5765-9_22 .
  • Construction of distributive Sperner plane stars that are homogeneous in two points. In: Journal of Geometry. Volume 16, issue 1, 1981, pp. 83-92, doi: 10.1007 / BF01917577 .
  • Affine Relatives. In: Results in Mathematics. Volume 12, Birkhäuser, Basel 1987, pp. 1–26, doi: 10.1007 / BF03322375 .
  • About a relational calculus for the algebraization of projective levels. In: Results in Mathematics. Volume 19, Birkhäuser, Basel 1991, pp. 211-233, doi: 10.1007 / BF03323282 .
  • The system concept of control theory and rule-relatives. In: Treatises from the Mathematical Seminar of the University of Hamburg. 28, Universität Berlin, Hamburg 1995, pp. 195-208, doi: 10.1007 / BF03322252 .

literature

Web links

Individual evidence

  1. ↑ H.-J. Arnold: Affine Relatives. In: Results in Mathematics. Volume 12, Birkhäuser, Basel 1987, pp. 1-26.
  2. ↑ H.-J. Arnold: A relation-theoretic algebraization of arranged affine and projective geometries. In: Treatises from the Mathematical Seminar of the University of Hamburg. Volume 45, Universität Berlin, Hamburg 1976, pp. 3-60.
  3. ↑ R. Soltysiak: The projection of affine structures over almost bodies with the help of relation-theoretic methods . Dissertation. University - Comprehensive University of Duisburg, 1980.
  4. ↑ A. Kopp: Development of relation-theoretical tools for algebraization and construction of general affine structures . Dissertation. University of Duisburg, 1986.
  5. ↑ CM Senevirathne: Relational characterization of semi-ordered affine and projective geometries . Dissertation. University - Comprehensive University of Duisburg, 1990.
  6. ↑ H.-J. Arnold: The system concept of control theory and rule-relatives. In: Treatises from the Mathematical Seminar of the University of Hamburg. Volume 28, University of Berlin, Hamburg 1995, pp. 195-208.
  7. ^ ED Sontag: Mathematical Control Theory. Deterministic Finite Dimensional Systems. 2nd edition, Springer-Verlag, 1998.
  8. ↑ P. Stemper: Relational construction of weakly affine geometries from linear control systems . Dissertation. University - Comprehensive University of Duisburg, 1997.
  9. ↑ M. Schleuter: Relational algebraic analysis of homogeneity properties in relatives established by control systems . Dissertation. University - Comprehensive University of Duisburg, 1997.
  10. ↑ D. Wetscheck: Fuzzification of control systems by means of relational algebraic and graph theoretical methods . Dissertation. University - Comprehensive University of Duisburg, 1999.
  11. ↑ A. Sauerland: Relative differential equations of classes of linear and non-linear control systems . Dissertation. University - Comprehensive University of Duisburg, 1994.
  12. ↑ H.-J. Arnold: About a relational calculus for the algebraization of projective levels. In: Results in Mathematics. Volume 19, Birkhäuser, Basel 1991, pp. 211-233.
  13. ↑ H.-J. Arnold: A remark on the homogeneity rule (H 2 × 2). (= Series of publications by the Department of Mathematics / Gerhard-Mercator-Universität Gesamtthoschulte Duisburg. Volume 370). 1997.
  14. ↑ H.-J. Arnold: Relational groupings in the context of Piaget's developmental psychology. In: Contributions to Geometric Algebra. 1977, pp. 361-366.
  15. ↑ E. Heineken, H.-J. Arnold, A. Kopp, R. Soltysiak: Strategies of thinking in the regulation of a simple dynamic system under different dead time conditions. In: Language & Cognition. 11/1986, pp. 136-148.
  16. ↑ H.-J. Arnold: For the mathematical description of goal-oriented human actions on technical systems. (= Series of publications by the Department of Mathematics / Gerhard-Mercator-Universität Gesamtthoschulte Duisburg. Volume 173). 1990.
  17. ↑ H.-J. Arnold: On the genesis of math in suitable fields of action. (= Series of publications by the Department of Mathematics / Gerhard-Mercator-Universität Gesamtthoschulte Duisburg. Volume 196). 1991.
personal data
SURNAME Arnold, Hans-Joachim
ALTERNATIVE NAMES Arnold, Hans-Joachim Felix (full name)
BRIEF DESCRIPTION German mathematician and university professor
DATE OF BIRTH March 31, 1932
PLACE OF BIRTH Berlin
DATE OF DEATH February 20, 2006
Place of death Mülheim an der Ruhr