A new simple Theory of Hypercomplex Integers

L. E. Dickson

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We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring, in order to extend the geometric approach of linear control to discrete event systems. We say that a subsemimodule of the free semimodule $𝒮^{n}$ over a semiring $𝒮$ is rational if it has a generating family that is a rational subset of $𝒮^{n}$ , $𝒮^{n}$ being thought of as a monoid under the entrywise product. We show that for various semirings of max-plus type whose elements are integers, rational...

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We consider the multiobjective decision making problem. The decision maker's (DM) impossibility to take consciously a preference or indifference attitude with regard to a pair of alternatives leads us to what we have called doubt attitude. So, the doubt may be revealed in a conscient way by the DM. However, it may appear in an inconscient way, revealing judgements about her/his attitudes which do not follow a certain logical reasoning. In this paper, doubt will be considered...

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This article formalizes the proof of Niven’s theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9].

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Eric Schmutz (2008)

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It is known that the unit sphere, centered at the origin in ℝn, has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point ν on the unit sphere in ℝn, and every ν > 0; there is a point r = (r 1; r 2;…;r n) such that: ⊎ ‖r-v‖∞ < ε.⊎ r is also a point on the unit sphere; Σ r i 2 = 1.⊎ r has rational coordinates; $r_{i} = \frac{a_{i}}{b_{i}}$ for some integers a i, b i.⊎ for all $i, 0 ⩽ |a_{i}| ⩽ b_{i} ⩽ {(\frac{32^{1 / 2} ⌈l o g_{2} n⌉}{ε})}^{2 ⌈l o g_{2} n⌉}$ . One consequence...