ARTICLE | doi:10.20944/preprints202112.0079.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Basic complex number theory; Euler’s identity; contradiction
Online: 6 December 2021 (15:02:06 CET)
In the paper it is demonstrated that a valid path to a contradiction in complex number theory exists. In the path use is made of Euler’s identity and simple trigonometry. Each step can be easily verified.
ARTICLE | doi:10.3390/sci2040072
Subject: Keywords: Yang–Baxter equation (QYBE); Euler’s formula; dual numbers; UJLA structures; classical means inequalities; poetry
Online: 21 October 2020 (00:00:00 CEST)
We consider a multitude of topics in mathematics where unification constructions play an important role: the Yang–Baxter equation and its modified version, Euler’s formula for dual numbers, means and their inequalities, topics in differential geometry, etc. It is interesting to observe that the idea of unification (unity and union) is also present in poetry. Moreover, Euler’s identity is a source of inspiration for the post-modern poets.
Subject: Mathematics & Computer Science, Other Keywords: Yang-Baxter equation; Euler’s formula; dual numbers; non-associative algebras; UJLA structures; mean inequalities
Online: 9 October 2019 (11:18:43 CEST)
This paper is a continuation of previous papers on unification theories publishes in AXIOMS. We present results about the (modified) Yang-Baxter equation, Euler’s formula, dual numbers, coalgebra structures, non-associative structures, differential geometry, and (mean) inequalities. We will also attempt to relate our discussion to some Brain Studies and Machine Learnig.
ARTICLE | doi:10.20944/preprints201610.0031.v1
Subject: Social Sciences, Finance Keywords: Solvency II; standard formula; risk measure; diversification; aggregation; monotony; homogeneity; subadditivity; Euler’s principle; capital allocation
Online: 10 October 2016 (11:53:00 CEST)
We introduce the notions of monotony, subadditivity, and homogeneity for functions defined on a convex cone, call functions with these properties diversification functions and obtain the respective properties for the risk aggregation given by such a function. Examples of diversification functions are given by seminorms, which are monotone on the convex cone of non-negative vectors. Any Lp norm has this property, and any scalar product given by a non-negative positive semidefinite matrix as well. In particular, the Standard Formula is a diversification function, hence a risk measure that preserves homogeneity, subadditivity, and convexity.