@article{Horn, author = {Horn, Martin Erik}, title = {If You Split Something into Two Parts, You Will Get Three Pieces: The Bilateral Binomial Theorem and its Consequences}, series = {Journal of Physics: Conference Series}, volume = {1612}, journal = {Journal of Physics: Conference Series}, number = {1}, publisher = {IOP Publishing}, issn = {1742-6588}, doi = {10.1088/1742-6596/1612/1/012013}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:523-15697}, pages = {1 -- 22}, abstract = {If a mathematical quantity is taken apart, the binomial expansion will result in three Pascal triangles. Thus a split into two parts generates a mathematical pattern of three pieces. This binomial expansion can now be generalized into a bilateral picture, resulting in an expansion of positive and negative powers as well. In so doing the Binomial Theorem will be generalized into the Bilateral Binomial Theorem, applying the intriguing mathematics of bilateral hypergeometric functions.And if a mathematical quantity is taken apart into two anti-commuting parts, the threefold pattern will triple again. Thus a ninefold symmetry appears. As an identical ninefold symmetry will appear if complex numbers and their conjugates are multiplied, complex conjugation must be seen as a strange, brutal, and illegitimate mathematical trick to model non-commutative structures by using commuting quantities.}, subject = {Binomischer Lehrsatz}, language = {en} } @article{Horn, author = {Horn, Martin Erik}, title = {Comments on the Kramer Kramer Man'ko 2 × 2 matrix multiplication}, series = {Journal of Physics: Conference Series}, volume = {1612}, journal = {Journal of Physics: Conference Series}, number = {1}, publisher = {IOP Publishing}, issn = {1742-6588}, doi = {10.1088/1742-6596/1612/1/012014}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:523-15702}, pages = {1 -- 13}, abstract = {This article is a complement to a paper by Kramer et al. that proposes a new matrix product, which is like a mixture of the usual and the Hadamard (or Schur) product. This new product is then related to the transformation group in the hyperbolic plane. Actually variations in the definition of internal operations lead to an infinity of products that can be associated with any of the Euclidean, hyperbolic or elliptic geometries in quite different manners. By the way: This article is not a review. The review can be found at the end of the paper.}, language = {en} }