Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals
We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix mul...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-12-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/10/4/340 |
| _version_ | 1797506601278504960 |
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| author | Wolf-Dieter Richter |
| author_facet | Wolf-Dieter Richter |
| author_sort | Wolf-Dieter Richter |
| collection | DOAJ |
| description | We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the <i>p</i>-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities. |
| first_indexed | 2024-03-10T04:34:53Z |
| format | Article |
| id | doaj.art-8e1eb87cf5bb49338d40eb0d36fd1887 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T04:34:53Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-8e1eb87cf5bb49338d40eb0d36fd18872023-11-23T03:50:20ZengMDPI AGAxioms2075-16802021-12-0110434010.3390/axioms10040340Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous FunctionalsWolf-Dieter Richter0Institute of Mathematics, University of Rostock, 18051 Rostock, GermanyWe generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the <i>p</i>-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities.https://www.mdpi.com/2075-1680/10/4/340vector multiplicationvector divisionvector exponential functiongeneralized trigonometric functionscomplex algebraic structuregeneralized complex plane |
| spellingShingle | Wolf-Dieter Richter Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals Axioms vector multiplication vector division vector exponential function generalized trigonometric functions complex algebraic structure generalized complex plane |
| title | Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals |
| title_full | Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals |
| title_fullStr | Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals |
| title_full_unstemmed | Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals |
| title_short | Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals |
| title_sort | complex numbers related to semi antinorms ellipses or matrix homogeneous functionals |
| topic | vector multiplication vector division vector exponential function generalized trigonometric functions complex algebraic structure generalized complex plane |
| url | https://www.mdpi.com/2075-1680/10/4/340 |
| work_keys_str_mv | AT wolfdieterrichter complexnumbersrelatedtosemiantinormsellipsesormatrixhomogeneousfunctionals |