This document discusses continued β-fractions in the field of Laurent series over a finite field Fq. Specifically:
1) It introduces the concept of β-expansions and continued β-fractions where the base β is a unit Pisot series in Fq((x-1)).
2) It summarizes previous work characterizing elements of Fq((x-1)) having finite β-fractions when β is a quadratic Pisot unit.
3) The main result of the paper is to improve upon this by studying the case when β is a Pisot unit (not necessarily quadratic) in Fq((x-1)).
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The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
https://utilitasmathematica.com/index.php/Index
Utilitas Mathematica journal that publishes original research. This journal publishes mainly in areas of pure and applied mathematics, statistics and others like algebra, analysis, geometry, topology, number theory, diffrential equations, operations research, mathematical physics, computer science,mathematical economics.And it is official publication of Utilitas Mathematica Academy, Canada.
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The sequence spaces 𝑙∞(𝑢,𝑣,Δ), 𝑐0(𝑢,𝑣,Δ) and 𝑐(𝑢,𝑣,Δ) were recently introduced. The matrix classes (𝑐 𝑢,𝑣,Δ :𝑐) and (𝑐 𝑢,𝑣,Δ :𝑙∞) were characterized. The object of this paper is to further determine the necessary and sufficient conditions on an infinite matrix to characterize the matrix classes (𝑐 𝑢,𝑣,Δ ∶𝑏𝑠) and (𝑐 𝑢,𝑣,Δ ∶ 𝑙𝑝). It is observed that the later characterizations are additions to the existing ones
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In this paper we define the generalized Cesaro sequence spaces 푐푒푠(푝, 푞, 푠). We prove the space 푐푒푠(푝, 푞, 푠) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map 푐푒푠 푝, 푞, 푠 to 푙∞ and 푐푒푠(푝, 푞, 푠) to c, where 푙∞ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
Matrix Transformations on Paranormed Sequence Spaces Related To De La Vallée-...inventionjournals
In this paper, we determine the necessary and sufficient conditions to characterize the matrices which transform paranormed sequence spaces into the spaces 푉휎 (휆) and 푉휎 ∞(휆) , where 푉휎 (휆) denotes the space of all (휎, 휆)-convergent sequences and 푉휎 ∞(휆) denotes the space of all (휎, 휆)-bounded sequences defined using the concept of de la Vallée-Pousin mean.
Fuzzy random variables and Kolomogrov’s important resultsinventionjournals
:In this paper an attempt is made to transform Kolomogrov Maximal inequality, Koronecker Lemma, Loeve’s Lemma and Kolomogrov’s strong law of large numbers for independent, identically distributive fuzzy Random variables. The applications of this results is extensive and could produce intensive insights on Fuzzy Random variables
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
https://utilitasmathematica.com/index.php/Index
Utilitas Mathematica journal that publishes original research. This journal publishes mainly in areas of pure and applied mathematics, statistics and others like algebra, analysis, geometry, topology, number theory, diffrential equations, operations research, mathematical physics, computer science,mathematical economics.And it is official publication of Utilitas Mathematica Academy, Canada.
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The study sheds light on the two-fuzzy normed space concentrating on some of their properties like convergence, continuity and the in order to study the relationship between these spaces
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In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
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1. *Corresponding Author: Rania Kammoun, Email: raniakammoun32@gmail.com
RESEARCH ARTICLE
Available Online at www.ajms.in
Asian Journal of Mathematical Sciences 2017; 1(6):230-233
Continued 𝜷-fractions with Pisot unit base in 𝑭𝒒 ((𝒙−𝟏))
Rania Kammoun*
* University of Sfax, Faculty of Sciences, Department of Mathematics, Algebra Laboratory, Geometry and Spectral
Theory (AGTS) LR11ES53, BP 802, 3038 Sfax, Tunisia.
Receivedon:15/11/2017,Revisedon:01/12/2017,Acceptedon:29/12/2017
ABSTRACT
In this paper, we are interested in introducing a new theory of continued fractions based on the beta-
expansion theory in the field of Laurent series over a finite field 𝐹𝑞. We will characterize all elements
having finite continued beta-fraction where the base is a unit Pisot quadratic series.
Classification Mathematic Subject: 11R06, 37B50.
Key words: Continued 𝛽-fraction, formal power series, Pisot series, 𝛽-expansion, finite field.
INTRODUCTION
The 𝛽-numeration introduced in 1957 by Rényi [5]
is a new numeration system when we replace the
integer base b with a non-integral base. Let 𝛽 > 1, in the case of a non-integral base, one may write any
𝑥 ∈ [0,1] as 𝑥 = ∑𝑘≥1
𝑥𝑘
𝛽𝑘 , where 𝑥𝑘 ∈ {0, ⋯ , [𝛽]}. The sequence (𝑥𝑘)𝑘≥1 is called an expansion of 𝑥 in
𝛽 base. There is no expansion uniqueness but, among them, the greatest sequence for the lexicographical
order is called the 𝛽-expansion of 𝑥and it is denoted by 𝑑𝛽(𝑥).
The 𝛽-expansion of 𝑥 is constructed by the greedy following algorithm. We consider the 𝛽-
transformation
𝑇𝛽: [0,1] → [0,1], 𝑥 → {𝛽𝑥} = 𝛽𝑥 − [𝛽𝑥]
and then we define
(𝑥𝑘)𝑘≥1 = 𝑑𝛽(𝑥) ≔ 𝑥1𝑥2𝑥3 ⋯, where 𝑥𝑘 = [𝛽𝑇𝛽
𝑘−1
(𝑥) ].
In the case 𝑥 ≥ 1, there exists a unique integer 𝑖 such that 𝛽𝑖−1
≤ 𝑥 < 𝛽𝑖
. So one can write
𝑥
𝛽𝑖 = ∑
𝑦𝑘
𝛽𝑘
𝑘≥1 ,
where (𝑦𝑘)𝑘≥1 is the 𝛽-expansion of
𝑥
𝛽𝑖 . Thus, we have
𝑥 = ∑ 𝑥𝑘𝛽−𝑘
∞
𝑘=−𝑛
𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥𝑘 = 𝑦𝑘−𝑛.
The 𝛽-integer part of 𝑥 is [𝑥]𝛽 = ∑ 𝑥𝑘𝛽−𝑘
∞
𝑘=−𝑛 and the 𝛽-fractional part of 𝑥is {𝑥}𝛽 = ∑ 𝑥𝑘𝛽−𝑘
𝑘>0 .
When {𝑥}𝛽 = 0, we denote by ℤ𝛽 the set of all 𝛽-integers.
Obviously, we can present an algorithm of continued fractions similarly to the classical decimal case by
consideration 𝛽 ∈ ℝ (non-integer) and then we get the so called continued 𝛽-fraction, whither the
sequence of partial quotients consists of 𝛽-integers instead of integers.
In [2]
, J. Bernat has showed that the continued 𝜙-fraction of 𝑥 is finite if and only if 𝑥 ∈ ℚ(𝜙). In [4]
, we
have studied the continued 𝛽-fraction with formal power series over finite fields and we have
characterize elements of 𝔽𝑞((𝑥−1
)) having finite 𝛽-fraction when the base 𝛽 is a quadratic Pisot unit.
Throughout this paper, we improve the result given [4]
by studying the case when 𝛽 is only a Pisot unit in
𝔽𝑞((𝑥−1
)). The paper is organized as follows, Section 2, we introduce some basic definitions and results.
In Section 3, we define the continued 𝛽-fraction expansion. In Section 4, we state our main result.