Eulers Totient Function (Number Theoretic Functions), Right Angle Triangle and Their Applications

In this paper, we introduce some new theorem and results(section Ⅱ,Ⅲ and Ⅵ) on Euler’ s Totient Function , Right angle triangle and their applications Keywords: Euler’ s totient function, right angle triangle, Golden ratio, Hardy – Ramanujan number, student’s pencil compass and etc.

FPGA-Implemented Fractal Decoder with Forward Error Correction in Short-Reach Optical Interconnects

Forward error correction (FEC) codes combined with high-order modulator formats, i.e., coded modulation (CM), are essential in optical communication networks to achieve highly efficient and reliable communication. The task of providing additional error control in the design of CM systems with high-performance requirements remains urgent. As an additional control of CM systems, we propose to use indivisible error detection codes based on a positional number system. In this work, we evaluated the indivisible code using the average probability method (APM) for the binary symmetric channel (BSC), which has the simplicity, versatility and reliability of the estimate, which is close to reality. The APM allows for evaluation and compares indivisible codes according to parameters of correct transmission, and detectable and undetectable errors. Indivisible codes allow for the end-to-end (E2E) control of the transmission and processing of information in digital systems and design devices with a regular structure and high speed. This study researched a fractal decoder device for additional error control, implemented in field-programmable gate array (FPGA) software with FEC for short-reach optical interconnects with multilevel pulse amplitude (PAM-M) modulated with Gray code mapping. Indivisible codes with natural redundancy require far fewer hardware costs to develop and implement encoding and decoding devices with a sufficiently high error detection efficiency. We achieved a reduction in hardware costs for a fractal decoder by using the fractal property of the indivisible code from 10% to 30% for different n while receiving the reciprocal of the golden ratio.

On the Eventually Periodic Continued β-Fractions and Their Lévy Constants

In this paper, we consider continued β-fractions with golden ratio base β. We show that if the continued β-fraction expansion of a non-negative real number is eventually periodic, then it is the root of a quadratic irreducible polynomial with the coefficients in Z[β] and we conjecture the converse is false, which is different from Lagrange’s theorem for the regular continued fractions. We prove that the set of Lévy constants of the points with eventually periodic continued β-fraction expansion is dense in [c, +∞), where c=12logβ+2−5β+12.

The Principle of Constitutional Identity and Individual Rights: In Search of the Golden Ratio

The phenomenon of the constitutional identity of the Russian state in the context of conventional and constitutional conflicts is currently acquiring a special meaning. The need to protect state sovereignty, upholding the traditional model of relations between the individual, society and the state are the urgent tasks facing the Russian Federation being the main focus of the constitutional reform of 2020. In this context, the purpose of the study was the legal analysis of the constitutional identity of the Russian state as a frontier of compromise between the European and national systems of protection of individual rights and freedoms. Disregard for the constitutional identity of the state may lead to its "erosion" and the adverse impact of internationalisation on the foundations of the constitutional order. Guided by the principle of constitutional identity, a state can arrive at the best and legitimate solution in a particular case, which would not contradict national law, but at the same time would not ignore the norms of international treaties. The application of both general scientific and special methods of cognition, the formal-legal method and the method of legal modelling, made it possible to see in the constitutional identity not only the boundary of compromises in Russia's relations with inter-state bodies, but also the distinctiveness of constitutional processes. The conducted analysis of the domestic legal acts led to the conclusion that the principle of constitutional identity is an inviolable foundation of the constitutional state and its model of ensuring the individual rights and freedoms in the traditional system of values. The indicators and trends of crime for the period from 1991 to 2020 are given.

Wavelet introscopy of human organism bionets

In domestic and foreign practice, a great deal of experience has been accumulated in the creation of means for monitoring the functional state of the human body. The existing complexes mainly analyze the electrocardiogram, blood pressure and a number of other physiological parameters. Diagnostics is often based on formal statistical data which are not always correct due to the nonstationarity of bioprocesses and without taking into account their physical nature. An urgent task of monitoring the state of the cardiovascular system is the creation of effective algorithms for computer technologies to process biosignals based on nonlinear dynamic models of body systems since biosystems and bioprocesses have a nonlinear nature and fractal structure. The nervous and muscular systems of the heart, the vascular and bronchial systems of the human body are examples of such structures. The connection of body systems with their organization in the form of self-similar fractal structures with scaling close to the “golden ratio” makes it possible to diagnose them topically. It is possible to obtain detailed information about the state of the human body’s bio-networks for topical diagnostics on the basis of the wavelet analysis of biosignals (the so-called wavelet-introscopy). With the help of wavelet transform, it is possible to reveal the structure of biosystems and bioprocesses, as a picture of the lines of local extrema of wavelet diagrams of biosignals. Mathematical models and software for wavelet introscopy make it possible to extract additional information from biosignals about the state of biosystems. Early detection of latent forms of diseases using wavelet introscopy can shorten the cure time and reduce the consequences of disorders of the functional state of the body (FSO), and reduce the risk of disability. Taking into account the factors of organizing the body’s biosystems in the form of self-similar fractal structures with a scaling close to the “golden ratio” makes it possible to create a technique for topical diagnostics of the most important biosystems of the human body.

A Novel LSB Matching Algorithm Based on Information Pre-Processing

This paper analyzes random bits and scanned documents, two forms of secret data. The secret data were pre-processed by halftone, quadtree, and S-Box transformations, and the size of the scanned document was reduced by 8.11 times. A novel LSB matching algorithm with low distortion was proposed for the embedding step. The golden ratio was firstly applied to find the optimal embedding position and was used to design the matching function. Both theory and experiment have demonstrated that our study presented a good trade-off between high capacity and low distortion and is superior to other related schemes.

On two relations characterizing the golden ratio

V.G. Zhuravlev found two relations associated with the golden ratio: $\tau=\frac{1+\sqrt{5}}{2}$: $[([i\tau]+1)\tau]=[i\tau^2]+1$ and $[[i\tau]\tau]+1=[i\tau^2]$. We give a new elementary proof of these relations and show that they give a characterization of the golden ratio. Further we consider satisfability of our relations for finite sets of $i$-s and establish some forcing property for this situation.