The Golden ratio has played an important role in musical composition, architecture, visual art, science, and increasingly in signal processing [1,2,3]. Underlying many of these applications are several extensions of the golden proportions including the Golden p-Section by Stakhov, the generalized Golden section by Bradley, and others [4,5]. In this paper we review and introduce generalizations of the Golden ratio. We show that there exists a fundamental connection between the limit of two consecutive terms of recursive sequences, the generalized (p, q)-Golden ratio and the Golden ratio generated by the characteristic equation. We apply these generalizations to forecasting financial time series to illustrate one of their applications in signal processing.
The present paper relates to the methods for data encoding and the reading of coded information represented by colored (including monochrome/black, gray) symbols (bars, triangles, circles, or other symbols). It also introduces new algorithms for generating secure, reliable, and high capacity color barcodes by using so called weighted n-dimensional random Fibonacci number based representations of data. The representation, symbols, and colors can be used as encryption keys that can be encoded into barcodes, thus eliminating the direct dependence on cryptographic techniques. To supply an extra layer of security, one may encrypt given data using different types of encryption methods.
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