The number of oriented rational links with a given deficiency number

Let [Formula: see text] be the set of un-oriented and rational links with crossing number [Formula: see text], a precise formula for [Formula: see text] was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let [Formula: see text] be the set of oriented rational links with crossing number [Formula: see text] and let [Formula: see text] be the set of oriented rational links with crossing number [Formula: see text] ([Formula: see text]) and deficiency [Formula: see text]. In this paper, we derive precise formulas for [Formula: see text] and [Formula: see text] for any given [Formula: see text] and [Formula: see text] and show that [Formula: see text] where [Formula: see text] is the [Formula: see text]th convolution of the convolved Fibonacci sequences.

On Third-Order Bronze Fibonacci Numbers

In this study, we firstly obtain De Moivre-type identities for the second-order Bronze Fibonacci sequences. Next, we construct and define the third-order Bronze Fibonacci, third-order Bronze Lucas and modified third-order Bronze Fibonacci sequences. Then, we define the generalized third-order Bronze Fibonacci sequence and calculate the De Moivre-type identities for these sequences. Moreover, we find the generating functions, Binet’s formulas, Cassini’s identities and matrix representations of these sequences and examine some interesting identities related to the third-order Bronze Fibonacci sequences. Finally, we present an encryption and decryption application that uses our obtained results and we present an illustrative example.

Steganography Technique and Modification of Substitution Cipher Using ASCII Code and Fibonacci Sequences

<br /><em>Cryptography is widely used to secure data and information so that it is not easily misused by parties who are not interested in the data. One type of cryptographic algorithm is Caesar Cipher and Vigenere Cipher. Both of these algorithms are classic cryptographic algorithms that need to be modified so that they become more optimal in the data security process. The first modification is to modify the Vigenere key using Fibonacci. In general, Vigenere Cipher will repeat the same keyword to complete the number of characters that are lacking so that the number of characters is the same as the number of characters in the plaintext. The second modification is to change or convert plaintext letters into ASCII letters so that the code is more difficult to solve. After the ASCII conversion process is done, the next results will be converted back in Hexa letters. In addition to the two modifications made, the steganography technique is also added by hiding the code behind the media in the form of images. Images that are sampled will be renamed and stored in different places.</em>

Generalized Finite-Length Fibonacci Sequences in Healthy and Pathological Human Walking: Comprehensively Assessing Recursivity, Asymmetry, Consistency, Self-Similarity, and Variability of Gaits

Healthy and pathological human walking are here interpreted, from a temporal point of view, by means of dynamics-on-graph concepts and generalized finite-length Fibonacci sequences. Such sequences, in their most general definition, concern two sets of eight specific time intervals for the newly defined composite gait cycle, which involves two specific couples of overlapping (left and right) gait cycles. The role of the golden ratio, whose occurrence has been experimentally found in the recent literature, is accordingly characterized, without resorting to complex tools from linear algebra. Gait recursivity, self-similarity, and asymmetry (including double support sub-phase consistency) are comprehensively captured. A new gait index, named Φ-bonacci gait number, and a new related experimental conjecture—concerning the position of the foot relative to the tibia—are concurrently proposed. Experimental results on healthy or pathological gaits support the theoretical derivations.

A note on Q-matrices and higher order Fibonacci polynomials

The results described in a recent article, relative to a representation formula for the generalized Fibonacci sequences in terms of Q-matrices are extended to the case of Fibonacci, Tribonacci and R-bonacci polynomials.

Representation of integers by k-generalized Fibonacci sequences and applications in cryptography

The aim of this paper is to construct a relation between tribonacci numbers and generalized tribonacci numbers. Besides, certain conditions are obtained to generalize the representation of a positive integer [Formula: see text] which is determined in [S. Badidja and A. Boudaoud, Representation of positive integers as a sum of distinct tribonacci numbers, J. Math. Statistic. 13 (2017) 57–61] for a [Formula: see text]-generalized Fibonacci numbers [Formula: see text]. Lastly, some applications to cryptography are given by using [Formula: see text].

Fault Prediction using a Grey-Markov Model from the Dissolved Gases Contents in Transformer Oils

A novel method to predict transformer fault by forecasting the variation trend of the dissolved gases content is proposed. After the content of each feature gas, such as hydrogen and methane, is obtained by the proposed forecasting model, the fault type can be diagnosed by the dissolved gas analysis (DGA) technologies. Firstly, the GM (1,1) grey model with unequal time interval is introduced to generate a general forecasting model for each feature gas. The introduced grey model with unequal time interval will enforce no constrain on the historical measurement data. Consequently, the time intervals of the two adjacent measuring points can be either constant or variant. To address the deficiency that the existing grey model is unable to describe the fluctuation of the predicted object in time domain, the Markov chain is introduced to improve the accuracy of the grey forecasting model. An adaptive method to automatically divide the state space based on the number of states and the relative error of the grey model is presented by using Fibonacci sequences. Practical measurements are used to verify the accuracy of the proposed forecasting model. The numerical results show that there is high probability (86%) that the proposed grey-Markov model acquires a smaller prediction residual as compared to the original GM(1,1) grey model.