DSpace Kolekcja: http://hdl.handle.net/11320/3539 2020-11-25T05:07:39Z Epsilon Numbers and Cantor Normal Form http://hdl.handle.net/11320/3552 Tytu&#322;: Epsilon Numbers and Cantor Normal Form Autorzy: Bancerek, Grzegorz Abstrakt: An epsilon number is a transfinite number which is a fixed point of an exponential map: ωϵ = ϵ. The formalization of the concept is done with use of the tetration of ordinals (Knuth's arrow notation, ↑). Namely, the ordinal indexing of epsilon numbers is defined as follows: and for limit ordinal λ: Tetration stabilizes at ω: Every ordinal number α can be uniquely written as where κ is a natural number, n1, n2, …, nk are positive integers, and β1 > β2 > … > βκ are ordinal numbers (βκ = 0). This decomposition of α is called the Cantor Normal Form of α. 2009-01-01T00:00:00Z Basic Properties of Periodic Functions http://hdl.handle.net/11320/3551 Tytu&#322;: Basic Properties of Periodic Functions Autorzy: Li, Bo; Li, Dailu; Men, Yanhong; Liang, Xiquan Abstrakt: In this article we present definitions, basic properties and some examples of periodic functions according to . 2009-01-01T00:00:00Z Complex Integral http://hdl.handle.net/11320/3549 Tytu&#322;: Complex Integral Autorzy: Yamazaki, Masahiko; Yamazaki, Hiroshi; Shidama, Yasunari; Wasaki, Katsumi Abstrakt: In this article, we defined complex curve and complex integral. Then we have proved the linearity for the complex integral. Furthermore, we have proved complex integral of complex curve's connection is the sum of each complex integral of individual complex curve. 2009-01-01T00:00:00Z On the Lattice of Intervals and Rough Sets http://hdl.handle.net/11320/3550 Tytu&#322;: On the Lattice of Intervals and Rough Sets Autorzy: Grabowski, Adam; Jastrzębska, Magdalena Abstrakt: Rough sets, developed by Pawlak , are an important tool to describe a situation of incomplete or partially unknown information. One of the algebraic models deals with the pair of the upper and the lower approximation. Although usually the tolerance or the equivalence relation is taken into account when considering a rough set, here we rather concentrate on the model with the pair of two definable sets, hence we are close to the notion of an interval set. In this article, the lattices of rough sets and intervals are formalized. This paper, being essentially the continuation of , is also a step towards the formalization of the algebraic theory of rough sets, as in  or . 2009-01-01T00:00:00Z