Sequences—Basic elements for discrete mathematics

In green wireless communication, the wireless network is preferred to design a system for low power consumption to minimize the environmental impacts of devices. An antenna is an essential tool in the transceiver system. In a multi-user network environment, large antenna arrays are required that consume high power. This paper proposes a solution using a thinned array antenna (TAA) to minimize energy use by the antenna module. For the array thinning, linear mathematical sequences like the prime number sequence and the Fibonacci sequence are used instead of conventional optimization methods. The antennas are arranged according to these sequences, and the performances are compared with the corresponding fully populated arrays. For a thinned array of 51 elements, using the prime number sequence 36.58% power saving is achieved. For a thinned array of 34 elements, using the Fibonacci sequence 26.5% power saving is achieved. The mathematical sequences (the array sequences) are directly inserted in the program of array thinning. This method doesn’t need any optimization algorithm for array thinning and hardware to implement it. The power consumption in hardware can be avoided. Therefore, the proposed thinned array antenna design using mathematical sequences provides a low-power antenna system for green wireless communication.

In this paper, we analyze properties of prime number sequences produced by the alternating sum of higher-order subsequences of the primes. We also introduce a new sieve which will generate these prime number sequences via the systematic selection and elimination of prime number indices on the real number line.

Euclid's proof that there are infinitely many primes [3, p. 25] can be modified to yield a proof of a simple inequality: If P1, P2'... is the sequence of prime numbers and n ? 2 then PI P2. Pn > P,l+?I In 1907, Bonse [I] gave an elementary proof of a stronger inequality, now called Bonse's Inequality [4]: If n ? 4 then PI P2 . IN > Pn + I Bonse then used his inequality to prove that 30 is the largest integer mi with the following property:

A sequence of prime numbers {p, 2p+1, 4p+3,...,2n-1(p+1)-1} is called a Cunningham chain. These are finite sequences of prime numbers, for which each element is a Sophie Germain prime. It is conjectured that there are arbitrarily large such Cunningham chains, and these chains form an essential part of the study of Sophie Germain primes. In this paper, we aim to significantly improve existing bounds for the length of Cunningham chains by considering their behaviour in the framework of what we will define as rogueness.

The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ-Fibonacci statistical convergence, strong Δ-Fibonacci summability, and Δ-Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers.

In this paper, we have determined the limit of ratio of (n+1)th term to the nth term of famous sequences in mathematics like Fibonacci Sequence, Fibonacci – Like Sequence, Pell's Sequence, Generalized Fibonacci Sequence, Padovan Sequence, Generalized Padovan Sequence, Narayana Sequence, Generalized Narayana Sequence, Generalized Recurrence Relations of Fibonacci – Type sequence, Polygonal Numbers, Catalan Sequence, Cayley numbers, Harmonic Numbers and Partition Numbers. We define this ratio as limiting ratio of the corresponding sequence. Sixteen different classes of special sequences are considered in this paper and we have determined the limiting ratios for each one of them. In particular, we have shown that the limiting ratios of Fibonacci sequence and Fibonacci – Like sequence is the fascinating real number called Golden Ratio which is 1.618 approximately. We have shown that the limiting ratio of Pell's sequence is a real number called Silver Ratio and the limiting ratios for generalized Fibonacci sequence are metallic ratios. We have also obtained the limiting ratios of Padovan and generalized Padovan sequence. The limiting ratio of Narayana sequence happens to be a number called super Golden Ratio which is 1.4655 approximately. We have shown that the limiting ratios of Generalized Narayana sequence are the numbers known as super Metallic Ratios. We have also shown that the limiting ratio of generalized recurrence relation of Fibonacci type is 2 and that of Polygonal numbers and Harmonic numbers are 1. We have proved that the limiting ratio of the famous Catalan sequence and Cayley numbers are 4. Finally, assuming Rademacher's Formula, we have shown that the limiting ratio of Partition numbers is the natural logarithmic base e. We have proved fourteen theorems to derive limiting ratios of various well known sequences in this paper. From these limiting ratio values, we can understand the asymptotic behavior of the terms of all these amusing sequences of numbers in mathematics. The limiting ratio values also provide an opportunity to apply in lots of counting and practical problems.

This study aims to reveal mathematical connections of elementary school teachers in solving trigonometric problem. The subjects of this study were 22 elementary school teachers as the prospective participants of Professional Teacher Education and Training (PTET). They came from several districts of South Sulawesi Province. The teachers were given trigonometry problem. Trigonometry problems could encourage teachers to connect geometrical and algebraic concept, graphical representation and algebraic representation, as well as daily life context. The result shows that most of the subject teachers of this study solved the problem according to procedures they know without considering everyday life context. On the other hand, there were some subjects who connected problem with everyday life context using graphical, verbal, or numerical representation. Thus, subjects who were able to connect problem information with appropriate concepts and procedures are categorized as substantive connections. While the subjects who were able to connect problem information with mathematical concepts but less precise in using the procedure are categorized as classification connections.

On the evolution and importance of the Fibonacci sequence in visualization of fractals

Prime Number Sequence Within Even Numbers

During the analysis of the fractal-primorial periodicity of the natural series of numbers, presented in the form of an alternation (sequence) of prime numbers (1 smallest prime factor > 1 of any integer), the regularity of prime numbers distribution was revealed. That is, the theorem is proved that for any integer = N on the segment of the natural series of numbers from 1 to N + 2N: (1) prime numbers are arranged in groups, by exactly three consecutive prime numbers of the form: (Р1-Р2-Р3). In this case, the distance from the first to the third prime number of any group is less than 2N integers, that is, Р3–Р1 < 2N integers. (2) These same prime numbers are redistributed in a line in groups, by exactly two consecutive prime numbers, on all segments of the natural series of numbers shorter than 2Nintegers.

Increasing Reality and Educational Merits of a Virtual Game

<em>The aim of this research is to analyze students' mathematical representation abilities in terms of cognitive style in the multivariable calculus course taught in semester 4 of the IKIP PGRI Pontianak Mathematics Education Study Program. This research is a qualitative descriptive research in the form of case study research. Research subjects were selected based on their cognitive style which was measured using the geft instrument. Selected subjects were then asked to work on mathematical representation ability questions and then interviewed according to these questions. Based on the research results, it was found that students who had a field dependent cognitive style in symbolic representation were able to link symbols with mathematical concepts, in graphical representation they were able to create images in the form of requested graphs, while in verbal representation they were less able to convey ideas effectively orally. or written. Meanwhile, students with an independent cognitive style in symbolic representation have been able to link symbols with mathematical concepts, in graphic representation they have been able to create images in the form of requested graphics, and in verbal representation they have been able to convey ideas effectively.</em>

The divergence and curl operators are fundamental mathematical tools introduced in university-level electromagnetism courses. Although students generally understand the operational use of these concepts, developing a clear physical intuition of what divergence and curl are can be challenging. This difficulty often arises due to the superficial or confusing treatment found in many standard textbooks. The goal of this article is to provide a clear demonstration, based on examples, that two-dimensional visualizations can be misleading when used to interpret these operators. A combination of graphical and analytical (symbolic) representations was found to be essential for developing an accurate physical intuition about divergence and curl. A deeper comprehension of these operators not only improves proficiency in electromagnetism, but also benefits a range of other fields, from fluid dynamics to meteorology and to technical applications.

We describe and analyze a situation involving symbolic representation and graphical visualization of the solution of a system of two linear differential equations, using a computer algebra system. Symbolic solution and graphical representation complement each other. Graphical representation helps to understand the behavior of the symbolic solution. Together with it, the symbolic solution and its analysis are needed to understand the graphical representation and to overcome the limiting constraints of the CAS. The study described in this paper points out the importance of flexibility in building connections between different mathematical registers, together with the fact that the balance between the usage of symbolic representations and of graphical representations can be very different from one student to the other.

We present an analysis of several characteristics of examples in 10 college algebra textbooks used in community colleges or 4-year institutions. We analyzed the examples along four dimensions: cognitive demand, the responses expected, the use of representations, and the strategies available for verifying the correctness of the solutions. We found that textbooks, independent of the type of institution in which they are used, present examples that have low cognitive demands, expect single numeric answers, emphasize symbolic and numerical representations, and give very few strategies for verifying correctness of the solutions. We discuss the implications of these findings for teaching and further research.