Some properties of integration of real-valued function over a fuzzy interval

One of the most fundamental concepts in fuzzy set theory is the extension principle. It gives a generic way of dealing with fuzzy quantities by extending non-fuzzy mathematical concepts. There are a few examples, including the concept of fuzzy distance between fuzzy sets. The extension approach is then methodically applied to real algebra, with considerable development of fuzzy number operations. These operations are computationally appealing and generalized interval analysis. Although the set of real fuzzy numbers with extended addition or multiplication is no longer a group, it retains many structural qualities. The extension concept is demonstrated to be particularly beneficial for defining set-theoretic operations for higher fuzzy sets. We need some definitions related to our properties before we can create the properties of integration of a crisp real-valued function over a fuzzy interval. It is our goal in this article to develop and demonstrate certain characteristics of a real-valued function over a fuzzy interval in order to broaden the scope of the notion of integration of a real-valued function over a fuzzy interval. Some of these characteristics are linked to the operations of extended addition and extended subtraction, while others are not.

On the norm-preservation of squares in real algebra representation

One of the main results of the article Gelfand theory for real Banach algebras, recently published in [Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM 114(4):163, 2020] is Proposition 4.1, which establishes that the norm inequality $$\Vert a^{2}\Vert \le \Vert a^{2}+b^{2}\Vert $$ ‖ a 2 ‖ ≤ ‖ a 2 + b 2 ‖ for $$a,b\in {\mathcal {A}}$$ a , b ∈ A is sufficient for a commutative real Banach algebra $${\mathcal {A}}$$ A with a unit to be isomorphic to the space $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) of continuous real-valued functions on a compact Hausdorff space $${\mathcal {K}}$$ K . Moreover, in this proposition is also shown that if the above condition (which involves all the operations of the algebra) holds, then the real-algebra isomorphism given by the Gelfand transform preserves the norm of squares. A very natural question springing from the above-mentioned result is whether an isomorphism of $${\mathcal {A}}$$ A onto $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) is always norm-preserving of squares. This note is devoted to providing a negative answer to this problem. To that end, we construct algebra norms on spaces $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) which are $$(1+\epsilon )$$ ( 1 + ϵ ) -equivalent to the sup-norm and with the norm of the identity function equal to 1, where the norm of every nonconstant function is different from the standard sup-norm. We also provide examples of two-dimensional normed real algebras $${\mathcal {A}}$$ A where $$\Vert a^2\Vert \le k \Vert a^2+b^2\Vert $$ ‖ a 2 ‖ ≤ k ‖ a 2 + b 2 ‖ for all $$a,b\in {\mathcal {A}}$$ a , b ∈ A , for some $$k>1$$ k > 1 , but the inequality fails for $$k=1$$ k = 1 .

Determinação aproximada de chuvas intensas no leste do paraná através da estatística e da álgebra pseudo real / Approximate determination of heavy rain in east paraná through statistics and pseudo real algebra

Este artigo foi elaborado para se mostrar como se pode utilizar a álgebra pseudo-real em correlações usando dados de precipitações intensas de localidades da região leste do Estado do Paraná, de forma a se obter equações de chuvas intensas aproximadas para uso em engenharia de recursos hídricos e aqui vamos analisar uma equação aproximada geral de chuvas intensas para esta região com dados de chuvas intensas obtidas no livro Chuvas Intensas no Brasil correlacionando-as através do uso da estatística pseudo real com as latitudes, as longitudes e as precipitações médias anuais obtidas pela internet.

Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

The Cayley–Klein (CK) formalism is applied to the real algebra so(5) by making use of four graded contraction parameters describing, in a unified setting, 81 Lie algebras, which cover the (anti-)de Sitter, Poincaré, Newtonian and Carrollian algebras. Starting with the Drinfel’d–Jimbo real Lie bialgebra for so(5) together with its Drinfel’d double structure, we obtain the corresponding CK bialgebra and the CK r-matrix coming from a Drinfel’d double. As a novelty, we construct the (first-order) noncommutative CK spaces of points, lines, 2-planes and 3-hyperplanes, studying their structural properties. By requiring dealing with real structures, we found that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we found 14 classical r-matrices coming from Drinfel’d doubles, obtaining new results for the de Sitter so(4,1) and anti-de Sitter so(3,2) as well as for some of their contractions. These geometric results were exhaustively applied onto the (3 + 1)D kinematical algebras, considering not only the usual (3 + 1)D spacetime but also the 6D space of lines. We established different assignations between the geometrical CK generators and the kinematical ones, which convey physical identifications for the CK contraction parameters in terms of the cosmological constant/curvature Λ and the speed of light c. We, finally, obtained four classes of kinematical r-matrices together with their noncommutative spacetimes and spaces of lines, comprising all κ-deformations as particular cases.

Correction: Bustamante et al. Determining When an Algebra Is an Evolution Algebra. Mathematics 2020, 8, 1349

The authors wish to make the following corrections to this paper [1] (see corrected version in postprint [2]):On page 2, paragraph 4, complete the first sentence ‘In Theorem 2 we show that if A is a real algebra and B is a basis of A then B also is a basis of Aℂ, the complexification of A (with the same multiplication structure matrices) and that A is an evolution algebra if, and only if, Aℂ is an evolution algebra’ with the phrase ‘and has a natural basis consisting of elements of A’ [...]

Pesquisas ambientais: o uso de álgebra pseudo real exata em ciências / Environmental research: the use of exact pseudo real algebra in cience

Este artigo foi feito para se mostrar que em matemática pseudo real podem existir raízes de números negativos, o que contraria a matemática tradicional, utilizando na construção das equações dos tipos potência e outras formulações formas hiperbólicas, com uns adendos logaritmos e exponenciais e exata, e isto na forma de uma análise ainda inicial que irá ser desenvolvida. Aqui só se mostra o início deste trabalho, que esperamos possamos desenvolver com o tempo para ser usado principalmente em pesquisas e análises ambientais.

Partial qualitative analysis of planar 𝓐Q-Riccati equations

If we view the field of complex numbers as a 2-dimensional commutative real algebra, we can consider the differential equation z'=az2+bz+c as a particular case of 𝓐- Riccati equations z'=a · (z · z)+b · z+c where 𝓐=( ℝn,·) is a commutative, possibly nonassociative algebra, a,b,c∈𝓐 and z:I → 𝓐 is defined on some nontrivial real interval. In the case 𝓐=ℂ, the nature of (at most two) critical points can be described using purely algebraic conditions involving involution * of ℂ. In the present paper we study the critical points of 𝓛(π)- Riccati equations, where 𝓛(π) is the limit case of the so-called family of planar Lyapunov algebras, which characterize 2-dimensional homogeneous systems of quadratic ODEs with stable origin. The number of possible critical points is 1, 3 or ∞, depending on coefficients. The nature of critical points is also completely described. Finally, simultaneous stability of the origin is considered for homogeneous quadratic part corresponding to algebras 𝓛(θ).

Planar Lyapunov Algebras

We study complex involutive algebras generated by a single nonselfadjoint idempotent and use them to construct a family of algebras, which we call planar Lyapunov algebras. As our main result, we prove that every 2-dimensional commutative real algebra whose homogeneous Riccati differential equation is stable at the origin must be isomorphic either to an algebra with zero multiplication or to some planar Lyapunov algebra.

Geometry of balanced and absorbing subsets of topological modules

We define the concepts of balanced set and absorbing set in modules over topological rings, which coincide with the usual concepts when restricting to topological vector spaces. We show that in a topological module over an absolute semi-valued ring whose invertibles approach [Formula: see text], every neighborhood of [Formula: see text] is absorbing. We also introduce the concept of total closed unit neighborhood of zero (total closed unit) and prove that the only total closed unit of the quaternions [Formula: see text] is its closed unit ball [Formula: see text]. On the other hand, we also prove that if [Formula: see text] is an absolute semi-valued unital real algebra, then its closed unit ball [Formula: see text] is a total closed unit. Finally, we study the geometry of modules via the extreme points and the internal points, showing that no internal point can be an extreme point and that absorbance is equivalent to having [Formula: see text] as an internal point.