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http://hdl.handle.net/11320/3539
Wed, 25 Nov 2020 05:31:04 GMT2020-11-25T05:31:04ZEpsilon Numbers and Cantor Normal Form
http://hdl.handle.net/11320/3552
Tytuł: 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.Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/11320/35522009-01-01T00:00:00ZBasic Properties of Periodic Functions
http://hdl.handle.net/11320/3551
Tytuł: 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 [5].Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/11320/35512009-01-01T00:00:00ZComplex Integral
http://hdl.handle.net/11320/3549
Tytuł: 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.Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/11320/35492009-01-01T00:00:00ZOn the Lattice of Intervals and Rough Sets
http://hdl.handle.net/11320/3550
Tytuł: On the Lattice of Intervals and Rough Sets
Autorzy: Grabowski, Adam; Jastrzębska, Magdalena
Abstrakt: Rough sets, developed by Pawlak [6], 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 [3], is also a step towards the formalization of the algebraic theory of rough sets, as in [4] or [9].Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/11320/35502009-01-01T00:00:00Z