An elementary unified approach to prove some identities involving Fibonacci and Lucas numbers

Using the explicit formulas of the generating polynomials of Fibonacci and Lucas, we prove some new identities involving Fibonacci and Lucas numbers. As an application of these identities, we show how some Diophantine equations have infinitely many solutions. To illustrate the powerful of this elementary method, we give proofs of many known formulas.

Categorization of Bridges by Failure Consequences

Categorization of bridge constructions by the failure consequences due to loss of ultimate capacity or serviceability is a difficult task that can be resolved using two different approaches: elementary method using currently valid standards advanced method based on risk assessment. The elementary method is grounded on subjective assessment of bridge malfunctioning or collapse. The bridge is classified into an appropriate category (consequence class) in accordance with the most severe consequence. It is a simple procedure that may be applied without the need of demanding mathematical procedures. However, the resulting categorization may be affected by uncertainties in the assessment of consequences. The advanced method is based on a comprehensive analysis of bridge malfunctioning using a procedure of risk assessment. This method takes account of the occurrence probability of unfavourable events and the significance of individual consequences. Classification of a bridge construction into an appropriate category depends on the resulting risk of the bridge failure. This more laborious approach provides credible results without excessive uncertainties. The most adverse difficulty of the method is a combination of some consequences like the loss of life and economic costs. Both the above-mentioned approaches to the categorization of bridges by failure consequences can be effectively used depending on the type of bridge, on the intensity of exploitation, on expected consequences, and on social impacts of bridge malfunctioning.

Odd/even cube-full numbers

In this paper we use an elementary method to give an asymptotical ratio of odd to even cube-full numbers and show that it is asymptotically 1 : 1 + 2^{−1/3} + 2^{−2/3}.

A direct computation of a certain family of integrals

PurposeThe authors propose a rather elementary method to compute a family of integrals on the half line, involving positive powers of sin x and negative powers of x, depending on the integer parameters n≥q≥1.Design/methodology/approachCombinatorics, sine and cosine integral functions.FindingsThe authors prove an explicit formula to evaluate sinc-type integrals.Originality/valueThe proof is not present in the current literature, and it could be of interest for a large audience.

Some new identities involving Laguerre polynomials

<abstract><p>In this paper, we use elementary method and some sort of a counting argument to show the equality of two expressions. That is, let $ f(n) $ and $ g(n) $ be two functions, $ k $ be any positive integer. Then $ f(n) = \sum\limits_{r = 0}^n(-1)^r\cdot \frac{n!}{r!}\cdot \binom{n+k-1}{r+k-1}\cdot g(r) $ if and only if $ g(n) = \sum\limits_{r = 0}^n(-1)^r\cdot \frac{n!}{r!}\cdot \binom{n+k-1}{r+k-1}\cdot f(r) $ for all integers $ n\geq0 $. As an application of this formula, we obtain some new identities involving the famous Laguerre polynomials.</p></abstract>

A study on optimally constructed compactly supported orthogonal wavelet filters

Compactly supported orthogonal wavelet filters are extensively applied to the analysis and description of abrupt signals in fields such as multimedia. Based on the application of an elementary method for compactly supported orthogonal wavelet filters and the construction of a system of nonlinear equations for filter coefficients, we design compactly supported orthogonal wavelet filters, in which both the scaling and wavelet functions have many vanishing moments, by approximately solving the system of nonlinear equations. However, when solving such a system about filter coefficients of compactly supported wavelets, the most widely used method, the Newton Iteration method, cannot converge to the solution if the selected initial value is not near the exact solution. For such, we propose optimization algorithms for the Gauss-Newton type method that expand the selection range of initial values. The proposed method is optimal and promising when compared to other works, by analyzing the experimental results obtained in terms of accuracy, iteration times, solution speed, and complexity.

On bivariate Jain operators

<p style='text-indent:20px;'>In this paper we deal with bivariate extension of Jain operators. Using elementary method, we show that these opearators are non-increasing in <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> when the attached function is convex. Moreover, we demonstrate that these operators preserve the properties of modulus of continuity. Finally, we present a Voronovskaja type theorem for the sequence of bivariate Jain operators.</p>

Quantum Cryptography: The Future of Internet and Security Management

In today’s world, cryptography techniques are used and implemented on the elementary method of finding the prime factor of large integer, which is said to be “Inevitable to Track”. But living in an era where nothing is impossible to achieve, so cryptographic techniques are exposed to both technologies’ advancement in computational power of machines and advancement in the fields of mathematics to break the notion that factoring of large integers into their prime is impossible. To cope up with the threat that cryptography will face is handled by fusion of physics into cryptography, leading to the evolution of Quantum Cryptography. It is one of the fast-growing fields in computer technology. In this paper, I am going to brief the concepts of Quantum Cryptography and how this technology has led to the development of the strategy of complete secure key distribution. The paper covers the loophole present in the modern cryptography techniques, the fundamental principle of quantum cryptography, its implementation in the real world along with the limitation faced in this field, and the possible future of quantum cryptography.

Evaluation of football players' actions in individual risky situations of offensive play as a model determinant of a sports game

The essence of effective training in a sports game is the use of means and forms of teaching in the organisation of a game, which most faithfully reflect the environment of the player's actual activity in natural conditions. Thus, there is a need to accurately identify the conditions of competition in a sports game. One of the types of such activities is observation, which being an elementary method of cognition, based on deliberate, targeted and intended as well as systematic perception of the studied subject, process or phenomenon. Research objective and questions The aim of the work is to assess the level of absolutely dependent cooperation (synergy) during a team sports game in changing game situations, in achieving the highest offensive objective, i.e. scoring a goal. - Is the effectiveness of achieving the highest objective of the game in offensive actions (i.e. scoring a goal) dependent on the level of player synergy? - Is the size of the conflict in the sports objective of the game dependent on creative player involvement? Material and test method - In the work, detailed observation was carried out with regard to players' actions in achieving the highest objective of the game, i.e. scoring a goal. The observation concerned analysis of 134 goals scored in 61 UEFA Champions League and European League football matches in 2009-2020. - In this study, the method of observation was used, with the possibility of repeatedly observing events being the subject of research. - In the research analysis, the use of film material of recorded games on a DVD was taken into account, where the observed effects were applied to an observation sheet. - The situations in which goals were scored and the research material comprised teams that competed in a given match. - The observation of scoring goals took the type of situation and level of interaction between players into account. Results Analysis of the obtained results can be inferred: 1. In achieving the highest objective of offensive actions, a high level of synergy prevails, requiring anticipation of events. 2. The high level of synergistic activities increases with the difficulty of situational tasks. 3. Due to the multi-faceted nature of team games, the player's training process should also affect his/her mental sphere.