Topological distances and geometry over the symmetrized Omega algebra

[EN] The aim of this paper is to study some topological distances properties, semidendrites and convexity on th symmetrized omega algebra. Furthermore, some properties and exponents on the symmetrized omega algebra are introduced.Alqahtani, M.; Özel, C.; Zekraoui, H. (2020). Topological distances and geometry over the symmetrized Omega algebra. Applied General Topology. 21(2):247-264. https://doi.org/10.4995/agt.2020.13049OJS247264212A. C. F. Bueno, On the exponential function of right circulant matrices, International Journal of Mathematics and Scientific Computing 3, no. 2 (2013).L. Hörmander, Notions of convexity, Progress in Mathematics 127, Birkh¨auser, Boston- Basel-Berlin (1994).S. Khalid Nauman, C. Ozel and H. Zekraoui, Abstract Omega algebra that subsumes min and max plus algebras, Turkish Journal of Mathematics and Computer Science 11 (2019) 1-10.G. L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: a brief introduction, Journal of Mathematical Sciences 140, no. 3 (2007), 426-444. https://doi.org/10.1007/s10958-007-0450-5D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161, American Mathematical Society, 2015. https://doi.org/10.1090/gsm/161C. Ozel, A. Piekosz, E. Wajch and H. Zekraoui, The minimizing vector theorem in symmetrized max-plus algebra, Journal of Convex Analysis 26, no. 2 (2019), 661-686.J.-E. Pin, Tropical semirings, Idempotency (Bristol, 1994), 50-69, Publ. Newton Inst., vol. 11, Cambridge Univ. Press, Cambridge, 1998. https://doi.org/10.1017/CBO9780511662508.004I. Simon, Recognizable sets with multiplicities in the tropical semiring, in: Mathematical Foundations of Computer Science (Carlsbad, 1988), Lecture Notes in Computer Science, vol. 324, Springer, Berlin, 1988, pp. 107-120. https://doi.org/10.1007/BFb001713

Some New Algebraic and Topological Properties of the Minkowski Inverse in the Minkowski Space

We introduce some new algebraic and topological properties of the Minkowski inverse A⊕ of an arbitrary matrix A∈Mm,n (including singular and rectangular) in a Minkowski space μ. Furthermore, we show that the Minkowski inverse A⊕ in a Minkowski space and the Moore-Penrose inverse A+ in a Hilbert space are different in many properties such as the existence, continuity, norm, and SVD. New conditions of the Minkowski inverse are also given. These conditions are related to the existence, continuity, and reverse order law. Finally, a new representation of the Minkowski inverse A⊕ is also derived

Comments on the fractional parts of Pisot numbers

summary:Let $L(\theta ,\lambda )$ be the set of limit points of the fractional parts $\lbrace \lambda \theta ^{n}\rbrace$, $n=0,1,2, \dots$, where $\theta$ is a Pisot number and $\lambda \in \mathbb{Q}(\theta )$. Using a description of $L(\theta ,\lambda )$, due to Dubickas, we show that there is a sequence $(\lambda _{n})_{n\ge 0}$ of elements of $\mathbb{Q}(\theta )$ such that $\operatorname{Card}\,(L(\theta ,\lambda _{n}))< \operatorname{Card}\,(L(\theta ,\lambda _{n+1}))$, $\forall$ $n\ge 0$. Also, we prove that the fractional parts of Pisot numbers, with a fixed degree greater than 1, are dense in the unit interval