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Certain matrices associated with balancing and Lucas-balancing numbers
No full textBalancing numbers and balancers are originally defined as the solution of the Diophantine equation . These numbers can be generated by the linear recurrence or by the nonlinear recurrence . There is another way to generated balancing numbers using powers of a matrix Q_{B} = begin{pmatrix} 6 & -1 \ 1 & 0\ end{pmatrix}. The matrix representation, indeed gives many known and new formulas for balancing numbers. In this paper, using matrix algebra we obtain several interesting results on balancing and related numbers
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Propriedades dos números balanceados de Lucas pelo método matricial
No full textOs números de balanceamento n e os balanceadores r são originalmente definidos como a solução do
Equação diofantina 1 + 2 + · · · + (n − 1) = (n + 1) + (n + 2) + · · · + (n + r). Se n é um balanceamento
número, então 8n 2 + 1 é um quadrado perfeito. Além disso, se n é um número de equilíbrio, então o positivo raiz quadrada de 8n
2+1 é chamado de número de equilíbrio de Lucas. Esses números podem ser gerados pelo recorrências lineares Bn+1 = 6Bn−Bn−1 e Cn+1 = 6Cn−Cn−1 onde Bn e Cn são respectivamente
denotado por no número de equilíbrio e no número de equilíbrio de Lucas. Existe outra maneira de
gerar números de balanceamento e balanceamento de Lucas usando potências de matrizes Q_B = (6 -1; 1 0) and Q_C = (17 -3; 3 -1) respectively. A representação matricial, de fato, fornece muitas fórmulas novas e conhecidas para balanceamento e Lucas-
números de equilíbrio. Neste artigo, usando álgebra matricial obtemos vários resultados interessantes sobre
Números de equilíbrio de Lucas.
Balancing numbers n and balancers r are originally dened as the solution of the Diophantine equation 1 + 2 + ... + (n - 1) = (n + 1) + (n + 2) + ... + (n + r). If n is a balancing number, then 8n^2 +1 is a perfect square. Further, If n is a balancing number then the positive square root of 8n^2 + 1 is called a Lucas-balancing number. These numbers can be generated by the linear recurrences B_n+1 = 6B_n - B_n-1 and C_n+1 = 6C_n - C_n-1 where B_n and C_n are respectively denoted by the nth balancing number and nth Lucas-balancing number. There is another way to generate balancing and Lucas-balancing numbers using powers of matrices Q_B = (6 -1; 1 0) and Q_C = (17 -3; 3 -1) respectively. The matrix representation, indeed gives many known and new formulas for balancing and Lucas-balancing numbers. In this paper, using matrix algebra we obtain several interesting results on Lucas-balancing numbers
A cryptography method based on hyperbolic balancing and Lucas-balancing functions
No full textThe goal is to study a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring balancing and Lucas-balancing numbers. These functions are indeed the extension of Binet formulas for both balancing and Lucas-balancing numbers in continuous domain. Some identities concerning hyperbolic balancing and Lucas-balancing functions are also established. Further, a new class of square matrices, a generalization of balancing QB-matrices for continuous domain, is considered. These matrices indeed enable us to develop a cryptography method for secrecy purpose
Certain Diophantine equations involving balancing and Lucas-balancing numbers
Get PDFIt is well known that if x is a balancing number, then the positive square root of 8x2 + 1 is a Lucas-balancing number. Thus, the totality of balancing number x and Lucas-balancing number y are seen to be the positive integral solutions of the Diophantine equation 8x2 +1 = y2. In this article, we consider some Diophantine equations involving balancing and Lucas-balancing numbers and study their solutions
Balancing and Cobalancing Numbers
Get PDFA different approach to the theory of balancing numbers is possible by means of a Pell’s equation which can be derived from the definition of balancing numbers. Each balancing number corresponds to a balancer, and each balancer, in turn, is a cobalancing number. Similarly, each cobalancing number corresponds to a cobalancer and interestingly, each cobalancer is a balancing number. The cobalancing numbers as well as their cobalancers are also solutions of a Diophantine equation similar to that satisfied by balancing numbers and their balancers. Some Diophantine equations exhibit beautiful solutions in terms of balancing and cobalancing numbers. The Lucas-balancing and Lucas-cobalancing numbers, obtained respectively as functions of balancing and cobalancing numbers, are useful in the computation of balancing and cobalancing numbers of higher order. Pell and associated Pell numbers are very closely associated with balancing and cobalancing numbers and appear in the factorization and as greatest common divisors of these numbers. The balancing, cobalancing and other related numbers are also expressible in terms of products of matrices
On the properties of k-balancing numbers
No full textAbstractIn this study, a generalization of the sequence of balancing numbers called as k-balancing numbers is considered and some of their properties are established. Further, the balancing polynomials that are the natural extension of k-balancing numbers are presented and observe that many of their properties admit straightforward proofs. The derivatives of these polynomials in the form of convolution are also discussed
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