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The first sentence in the abstract should be replaced with the sentence "A sequence $(x_{k})$ is called arithmetically convergent if for each $\varepsilon > 0$ there is an integer $n_{0}$ such that $|x_{m} - x_{}| \geq n_{0}$, where the symbol $< m, n >$ denotes the greatest common divisor of the integers $m$ and $n$".

O presente artigo discute como alunos de 6º ano do ensino fundamental lidam com uma tarefa de comparação de áreas de figuras planas em ambientes com características distintas: Papel e Lápis, Materiais Manipulativos e no software de geometria[1] Apprenti Géomètre 2. As bases teóricas da pesquisa são a Teoria dos Campos Conceituais de Gérard Vergnaud e a abordagem da área como grandeza, proposta por Régine Douady e Marie-Jeanne Perrin-Glorian. Os sujeitos da pesquisa resolveram a tarefa em duplas, em um dos três ambientes supracitados. Os ambientes Materiais Manipulativos e Apprenti Géomètre 2 proporcionaram uma maior diversidade de possibilidades de ação, em relação ao Papel e Lápis, como o decalque com papel manteiga, a decomposição efetiva das figuras com tesoura e as ferramentas do menu, tais como mover, rotacionar, decompor e agrupar, por meio das quais foi possível realizar decomposição e recomposição sem perda nem sobreposição e sobrepor as figuras. Entre os teoremas-em-ação verdadeiros mobilizados pelos sujeitos estão “a área é invariante por isometrias”, observado em todos os ambientes, e “figuras equidecompostas têm áreas iguais”, identificado na resolução com o Apprenti Géomètre 2. Também foram observados indícios de teoremas-em-ação falsos como “duas figuras que têm mesma área são congruentes”. [1] Mais informações sobre esse software podem ser encontradas em Silva (2016). Résumé Ce travail porte sur la résolution d’une tâche de comparaison d’aire de surfaces planes par des élèves de sixième, dans trois environnements différents: papier-crayon, matériels manipulatifs et le logiciel de géométrie Apprenti Géomètre 2. Son cadre théorique est celui de la Théorie des Champs Conceptuels de Gérard Vergnaud. L’approche de l’aire en tant que grandeur proposée par Régine Douady et Marie-Jeanne Perrin-Glorian est adoptée dans la recherche. Les sujets ont résolu la tâche en binômes, dans l’un des trois environnements cités. La possibilité d’utiliser le papier calque, les ciseaux et le scotch dans l’environnement matériels manipulatifs et les fonctionnalités du logiciel a permis une diversité plus grande de possibilités d’action, pour reproduire et déplacer les figures ainsi que pour faire des découpages-recollements sans perte ni chevauchement. Parmi les théorèmes-en-acte vrais mobilisés par les sujets il y a celui selon lequel l’aire est invariante par isométries, observé dans les trois environnements et celui selon lequel deux figures equidécomposables ont des aires égales, identifié dans le logiciel. Nous avons repéré aussi des signes de la mise en œuvre de théorèmes-en-acte faux comme, par exemple, deux figures qui ont même aire sont congruentes.

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Copyright of Educação Matemática Pesquisa is the property of Pontificia Universidade Catolica de Sao Paulo, Programa de Estudos Pos-Grad. em Educacao Matematica and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Educação Matemática Pesquisa is the property of Pontificia Universidade Catolica de Sao Paulo, Programa de Estudos Pos-Grad. em Educacao Matematica and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Educação Matemática Pesquisa is the property of Pontificia Universidade Catolica de Sao Paulo, Programa de Estudos Pos-Grad. em Educacao Matematica and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Lusitania Sacra is the property of Universidade Catolica Portuguesa, Centro de Estudos de Historia Religiosa and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let $\mathbb{F}_{q}$ denote the finite field of $q$ elements, where $q$ is a prime power. In this paper, we study the twisted Hessian curves denoted $H_{a,d}(\mathbb{F}_{q}[e])$ over the ring $\mathbb{F}_{q}[e]$, where $e^{3}=e^2$ and $(a,d)\in (\mathbb{F}_{q}[e])^{2}$. More precisely, we study some arithmetical properties of this ring and using the Twisted Hessian equation, we define the twisted Hessian curves $H_{a,d}(\mathbb{F}_{q}[e])$. This work study the twisted Hessian curve helped us to define two twisted Hessian over the finite field $\mathbb{F}_{q}$. We end this paper by giving the classification of the elements in $H_{a,d}(\mathbb{F}_{q}[e])$.

In this paper, a combination of boundary value problems for the nonhomogeneous analytic equations have been studied. The aim of this paper is to find a solution for the Schwarz-(Dirichlet-Neumann) problem and obtain its solvability conditions.

This paper presents several sufficient conditions for the existence of at least three classical solutions of a boundary value problem for a fourth-order difference equation. Fourth-order boundary value problems act as models for the bending or deforming of elastic beams. In different fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, economics and many others, the mathematical modelling of important questions leads naturally to the consideration of nonlinear difference equations. Our technical approach is based on variational methods. An example is included in the paper. Numerical computations of the example confirm our theoretical results.

With the current struggles of the world nowadays with several epidemics, modeling the dynamics of disease outbreaks has become much more important than any time before. In this context, the present paper aims at studying a stochastic hepatitis B virus epidemic model with treatment cure rate. Our model consists of three epidemic compartments describing the interaction between the susceptible, the infected and the recovered individuals; an SIR model where the infected individuals transmit the infection to the susceptible ones with a transmission rate perturbed by white noise. Our paper begins by establishing that our hepatitis B stochastic model has unique global solution. It moves then to giving sufficient conditions for the stochastic extinction and persistence of the hepatitis B disease. Finally, our paper provides some numerical results to support the analytical study, showing numerically that the treatment cure rate facilitates the extinction of the hepatitis B disease among the population.

In this paper, we are interested to prove a general fixed point theorem for a mapping in fuzzy b-metric spaces. The results in this paper generalize the Banach fixed point theorem in fuzzy b-metric spaces. To show the significance of our result an application is presented to establish the existence of a solution for an integral equation.

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Copyright of Educação Matemática Pesquisa is the property of Pontificia Universidade Catolica de Sao Paulo, Programa de Estudos Pos-Grad. em Educacao Matematica and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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Copyright of Revista Lusofona de Educacao is the property of Universidade Lusofona de Humanidades e Tecnologias, CEIEF and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper we propose an algebraic model for a modal epistemic logic. Although it is known the existence of algebraic models for modal logics, considering that there are so many different modal logics, so it is not usual to give an algebraic model for each such system. The basic epistemic logic used in the paper is bimodal and we can show that the epistemic algebra introduced in the paper is an adequate model for it.

This paper discusses the methodology for analysing historical mathematics textbooks. It is shown that the task of analysing mathematics textbooks of other epochs, and even of other cultures, constitutes challenging tasks, which have hitherto been largely ignored. The paper elaborates a broad programme for such analyses, developing it by using the textbook production by a prolific and highly successful French author as the case for showing how to work with this methodological programme. This author is Sylvestre-François Lacroix (1765-1843). Given that no text speaks by itself, the principal new elements introduced here for textbook analysis are categories for a socio-cultural contextualisation of textbook production. For structuring the work of analysing, a three-dimensional grid is proposed for realising such a contextualising. To avoid a schematic application, a number of concretising questions has been added, exploring the text and its context in the respective historical period. The grid and the questions are studied in this case for France, in the revealing period of establishing a public education system, and comparing and contrasting with Germany, and in particular with Prussia.

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Copyright of Aufklärung: Revista de Filosofia is the property of Aufklarung: Revista de Filosofia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Revista Científica Guillermo de Ockham is the property of Revista Cientifica Guillermo de Ockham, Universidad de San Buenaventura and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Revista Monografias Ambientais is the property of Revista Monografias Ambientais and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In an engine, the valve train is responsible for approximately 6% to 35% of the total friction losses, with the value varying with the design and operating conditions. Even a 1% improvement in the fuel economy of a vehicle model is of great economic and environmental relevance. One of the engine systems subject to considerable friction is the control system and actuation of gas admission and discharge valves, the valve control system. Friction efforts and wear between the cam and the tappet depend on the relative speeds of the parts, the state of the surfaces in contact, the lubrication and temperature conditions, and the materials and geometries of the cam and tappet. A carefully project of the cam-tappet system is needed to preserve parts integrity and system functionality. This work presents some important experiments carried out with various cam-tappet systems, measured on engines or constructed using parts of an engine. Several important quantities are measured and analyzed by the reported experiments. The main objective of this paper is to promote a better understanding of these quantities that can be useful to engine designers to minimize energy losses or for optimize desirable mechanical characteristics of the system. Also, this paper allows researchers to plan new experiments and/or rapidly identify typical experimental results. It was possible to conclude that typical friction coefficients between cam and tappet varies from 0.05 to 0.15, that the force between valve stem-valve guide represents only about 1.5 -2% of the force acting on the valve stem, that the cam-follower friction was low for viscous oil SAE 50, compared with other lubricants, whereas camshaft bearing friction was low for lubricant SAE 5W-40 in the same engine, that the friction at the cam-follower interface decreases with increasing engine speed, that viscous oil gave greater oil film thickness as compared to the low-viscosity lubricant at the cam-follower interface, among others.

Copyright of Colloquium Exactarum is the property of Asociacao Prudentina de Educacao e Cultura (APEC) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Colloquium Humanarum is the property of Asociacao Prudentina de Educacao e Cultura (APEC) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Em Extensao is the property of Em Extensao and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper aims to identify and analyze the theme of Financial Education (EF) in academic research in the latest editions of ENEM, CIBEM and SIPEM. The methodological approach is qualitative of the bibliographical. We believe that the SBEM electronic database enabled a bibliographic survey of academic production. The results pointed to the need to promote discussions about EF in the school environment, recognizing its contribution to the integral and critical formation of the subject, its articulation with social, political, economic, environmental and cultural problems, making education a progressively active social force. They also highlighted the urgent need to implement continuing education programs that enable the planning and implementation of more assertive teaching practices in this area.

Since the publication of the Base Nacional Comum Curricular (National Common Curricular Base), Financial Education has become a topic to be developed in the classroom, mainly related to the mathematics curriculum. In this sense, teachers are in charge of addressing it. In view of this, in this paper we aim to discuss understandings of Financial Education aimed at Basic Education and produced by mathematics undergraduates, mobilized in two training meetings. Based on Critical Mathematics Education and the qualitative research paradigm, we identified five topics using content analysis procedures. The results of the analysis of these topics indicate that the future teachers have knowledge of School Financial Education, although they relate the subject to economic aspects and Financial Mathematics.

The object of this paper is to evaluate some finite double summations relations for the multivariable $A$-function using the summation of a double hypergeometric series. The formulas derived in this paper are most general in character, we also provide a few particular cases for derived summation formulas.

The aim of this paper is to introduce a new class of mappings called almost nonexpansive mappings in a $b$-metric space. Some characteristics of this class of mappings are discussed. Fixed point and common fixed point results for such mappings are obtained. An application to the Cauchy problem in a Banach space is also shown in this paper.

In this paper, we discuss the implementation and numerical aspects of the Matlab solver designed for the solution of Low Protein Model (LPD). In this paper, we study the existence and uniqueness of the weak solution and we try to write some codes in Matlab which are based on Euler’s Method and several technics of programmation. subject to an initial condition dM dt = µL − δM, (2) L(0) = 0 and M (0) = 0. (3) The code is based on Euler’s Method and several technics of programmation.

Let S be a commutative semiring with unity. In this paper, we introduce the weakly nilpotent graph of a commutative semiring. The weakly nilpotent graph of S, denoted by 􀀀w(S) is defined as an undirected simple graph whose vertices are S and two distinct vertices x and y are adjacent if and only if xy 2 N(S), where S = Sn f0g and N(S) is the set of all non-zero nilpotent elements of S. In this paper, we determine the diameter of weakly nilpotent graph of an Artinian semiring. We prove that if 􀀀w(S) is a forest, then 􀀀w(S) is a union of a star and some isolated vertices. We study the clique number, the chromatic number and the independence number of 􀀀w(S). Among other results, we show that for an Artinian semiring S, 􀀀w(S) is not a disjoint union of cycles or a unicyclic graph. For Artinian semirings, we determine diam(􀀀w(S)). Finally, we characterize all commutative semirings S for which 􀀀w(S) is a cycle, where 􀀀w(S) is the complement of the weakly nilpotent graph of S. Finally, we characterize all commutative semirings S for which Γw(S) is a cycle.

In this paper we establish in first the existence of strong optimal solutions of a control problem for dynamics driven by a linear forward-backward doubly stochastic differential equations of mean-field type (MF-FBDSDEs), with random coefficients and non linear functional cost. Moreover, we establish necessary as well as sufficient optimality conditions for this kind of control problem. In the second part of this paper, we establish necessary as well as sufficient optimality conditions for existence of both optimal relaxed control and optimal strict control for dynamics of nonlinear mean-field forward-backward doubly stochastic differential equations.

We show that integration by parts formulas based on Malliavin-Skorohod calculus techniques for additive processes help us to compute quantities like ${\E}(L_T h(L_T))$ for different suitable functions $h$ and different models for the cumulative loss process $L_T$. These quantities are important in Insurance and Finance. For example they appear in computing expected shortfall risk measures or stop-loss contracts. The formulas given in the present paper, obtained by simple proofs, generalize the formulas given in a recent paper by Hillairet, Jiao and Réveillac using Malliavin calculus techniques for the standard Poisson process, a particular case of additive process.

In this paper, we introduce a viscosity-type extragradient algorithm for finding a common point of the solution of a pseudomonotone equilibrium problem and a fixed point problem of an infinite family of multi-valued quasi-nonexpansive mappings in real Hilbert space. Using our algorithm, we state and prove a strong convergence result of our iteration sequences. An application to variational inequality problem was considered. Lastly, we give a numerical example of our main result. The result presented in this paper extends and complements some recent results in literature.

This paper introduces a class of rings called left nil zero semicommutative rings ( LNZS rings ), wherein a ring R is said to be LNZS if the left annihilator of every nilpotent element of R is an ideal of R. It is observed that reduced rings are LNZS but not the other way around. So, this paper provides some conditions for an LNZS ring to be reduced and among other results, it is proved that R is reduced if and only if the ring of upper triangular matrices over R is LNZS. Furthermore, it is shown that the polynomial ring over an LNZS may not be LNZS and so is the case of the skew polynomial over an LNZS ring. Therefore, this paper investigates the LNZS property over the polynomial extension and skew polynomial extension of an LNZS ring with some additional conditions.

Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{2^r\cdot7^s}-m\in \mathbb{Z}[x]$, where $m\neq \pm 1$ is a square free integer, $r$ and $s$ are two positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\not\equiv 1\md{4}$ and $\overline{m}\not\in\{\pm \overline{1},\pm \overline{18},\pm \overline{19}\} \md{49}$, then $K$ is monogenic. But if $r\geq 2$ and $m\equiv 1\md{16}$ or $s\geq 3$, $\overline{m}\in\{ \overline{1}, \overline{18}, -\overline{19}\} \md{49}$, and $\nu_7(m^6-1)\geq 4$, then $K$ is not monogenic. Some illustrating examples are given at the end of the paper.

The logic PM4N was introduced by Jean-Yves Beziau as a modal and 4-valued system. In this introductory paper, the author presented the system from a matrix logic with four values disposed in a Boolean algebra with a modal operator for the notion of necessity. From that matrix semantics, the paper shows some valid results and stresses some motivations of the matrix semantics and of the system PM4N. In this paper we develop some additional aspects of that logic and present two deductive systems for it, a simple system of tableaux and a sequent calculus.

Copyright of Revista Espacios is the property of Talleres de Impresos Oma and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper characterizes the use of qualitative data analysis software in doctoral theses about Mathematics Education developed in Brazil, from 2010 to 2019, to support the choices of these software from paths already followed by other researchers. For this, a mapping was made to identify the theses and the aspects related to the use of these software in the field. The investigation retrieved 651 doctoral theses available in the Brazilian Digital Library of Thesis and Dissertations in which only 31 of them compose the corpus of this paper. This number evidences the little use of these software in the analyzed area. In the corpus, it was observed the predominance of the qualitative approach and the identification of nine different software, being three the most used: Atlas.ti, NVivo and CHIC, in that order. The adoption of software, as described by the authors, is linked to their potential regarding categorization, organization, visualization, data handling, time optimization and analysis facilitation and the possibility of enabling exploration of large volumes and diversity of data.

In this paper, we investigate the tensions and interferences between philosophy and mathematics in the context of the emergence of pure mathematics in Prussia in the 19th century. In historiography the occurrence of an epistemological turn in mathematics in that century is well known. Such occurrence came to restructure the internal conceptions of its objects, in detriment of a quantitative notion and instead from a relational conception. We argue here that this change was facilitated, precisely, by the Kantian understanding that mathematics, by dealing with synthetic a priori judgments, would conceive itself as the foundation of thinkability. Our approach resides upon investigating how these notions were addressed and conceived of in the works of two authors: Jakob Fries, philosopher of science and mathematics at the beginning of the 19th century, and Hermann Grassmann, mathematician whose work would influence new conceptions by the second half of that century. Our research identifies, in the perspectives and practices of Fries and Grassmann, a conceptual unity, regarding the foundational reflection of mathematics from abstract structures of thinking. This finding reveal, on the other hand, the intervention of philosophical practices and references within the mathematical practice, producing tensions and encompassing a relevant role in the epistemological change in question. [ABSTRACT FROM AUTHOR]

In this paper, we present the results of a new approach to the splitting factor preconditioner, which is a preconditioner based on the Incomplete Cholesky factorization and the splitting preconditioner. In previous work for small linear programming problems, the preconditioner was applied in all iterations of the interior point method and compared with the splitting preconditioner also applied in all iterations. In this paper, we will do a hybrid approach, in which in the first iterations the preconditioner is the Incomplete Cholesky Factorization, and in the last iterations, the preconditioner used is the splitting factor preconditioner or the splitting preconditioner. The results obtained show that even in the hybrid approach, the splitting factor preconditioner achieved better performance.

The generalized Kronecker graph product was introduced by Figueroa-Centeno et al. (2008). Later, Baca et al. (2018) used it for obtaining solutions of the n-queens problem on larger boards from solutions on smaller boards. In this paper, we generalize the graph product and the recent results by Baca et al. (2018), obtaining a larger class of solutions by knowing solutions on lower size boards in advance. We finalize the paper stating a couple of conjectures regarding conditions for obtaining composite solutions via graph product.

The undoubted benefits and intrinsic virtues of mathematics should not blind us to potential collateral damage as the huge juggernaut of mathematics rolls over education and society towards a reshaped future. Because of its great power and influence across the whole of society we need to undertake an ethical audit. This paper conducts an ethical critique that considers four interrelated aspects of mathematics in education and society and their negative impacts. 1. The overvaluation of mathematics and the effects this power has in maintaining privileges. 2. The powerful negative impacts of mathematical studies have on many individual learners. 3. Visible but problematic applications of mathematics within society that are shielded from criticism because mathematics continues to be viewed as neutral. 4. The profound and performative effects of the hidden applications of mathematics that reformat society and change our everyday lives but remain unchecked in every sense. Most of the paper is devoted to revealing these damaging effects. The solutions proposed are for ethical awareness to be fostered in mathematical study at all levels, as well as challenging the widespread stereotype of mathematics as ethically neutral.

This paper aims to identify how historical information can contribute to teacher education regarding to the Golden Ratio content. Therefore, in the theoretical framework, the relevance of the teacher's investigative stance and the contributions that the use of the History of Mathematics provides for the training of this professional are discussed, as well as about some aspects of the Golden Ratio. For this purpose, an implementation was carried out that involved the use of the History of Mathematics, in order to work on the Golden Ratio content with 4th year students of the Mathematics Degree course at a public educational institution. In this paper, the focus was directed to the study of myths and truths concerning the Golden Ratio during the planning of the implementation, as well as to the discussions of the students later. The research is qualitative in nature, based on participant observation for data collection and mixed categorization, for analysis of what had been constituted. We concluded that the use of historical information about the Golden Ratio content led to critical manifestations, so that the research teachers and students in initial training had the opportunity to deconstruct certain beliefs impregnated and disseminated in history.

In this paper, we introduce and study a system of set-valued Cayley type inclusions involving Cayley operator and (H; )-monotone operator in real Banach spaces. We show that Cayley operator associated with the (H; )-monotone operator is Lipschitz type continuous. Using the proximal point operator technique, we have established a fixed point formulation for the system of set-valued Cayley type inclusions. Further, the existence and uniqueness of the approximate solution are proved. Moreover, we suggest an iterative algorithm for the system of set-valued Cayley type inclusions and discuss the strong convergence of the sequences generated by the proposed algorithm. Some examples are constructed to illustrate some concepts used in this paper.

In their paper [1], Shahzad Ahmad et al. given a characterization on any pure sextic number field Q(m1/6) with square-free integers m satisfying m 6 ±1 (mod 9) to have a power integral bases or do not. In this paper, for these results, we give a new easier proof than that given in [1]. We further investigate the cases m 1 (mod 4) independently to the satisfaction of m2 1 (mod 9), m 1 (mod 9), and the number fields defined by x2r3t −m, where r, t are two non-negative integers, and m is a square free integer are investigated. The proposed proofs are based on Dedekind’s criterion and on prime ideal factorization.

In this paper we establish Hyers-Ulam-Rassias stability of a generalized functional equation in fuzzy Banach spaces. The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-30.