Computing Bounds for General Randic Coindex of Sum Graphs

The physical and structural properties of molecular structure or graph such as boiling point, melting point, surface tension, or solubility are studied using topological index (TI). Topological index is a mathematical formula that can be applied to any graph which models some molecular structures. The various operations play an important role in graph theory such as joining, union, intersection, products, and subdivision. In this paper, we computed the bounds for general Randic coindex of F -sum graphs such as ( S -sum, R -sum, Q -sum, and T -sum) in the form of their factor graphs. At the end, results are illustrated by numerical table for the particular F -sum graphs.

Approximate Calculation Method for Noncentral t-Distribution Quantile

In the process of structural design and structural performance evaluation, the inference of the reliable life of the structure and the representative value of the material strength is a necessary work. The determination of material strength is the presumption of the quantile of the normal distribution, and the determination of the confidence level of the quantile of the normal distribution involves the noncentral t-distribution function. However, the calculation of the quantile is very complicated and is often provided in the form of a numerical table, which often involves multiparameter interpolation calculation, so it is not convenient to apply. The existing approximate calculation methods for noncentral t-distribution quantiles have strict application conditions, and the calculation process is relatively cumbersome. It is still difficult to meet actual needs in terms of fitting accuracy, application range, and convenience. In this paper, a new calculation method for noncentral t-distribution quantiles is proposed by introducing new probability expressions and related approximate distributions, based on theoretical derivation and numerical fitting. The comparative analysis results show that the method not only is convenient for calculation but also has the advantages of higher accuracy and wider application range, and it is more in line with the actual needs of engineering.

Arithmetic Table as an Integral Part of all Computational Mathematics

The paper is devoted to studying the following issue as a statement. What do we know and what we don’t know about arithmetic tables. Perhaps there is no mathematical problem as naive or simple as finding a method for creating arithmetic tables. We confirm that the general method has not been found yet. This study provides nonterminal solution to this problem. Why? The presentation of arithmetic material in essence, plus some accompanying ideas, makes it possible to develop them further in the system. Materials and methods. The system looks like this: a numerical table as a Pascal's triangle and a symmetric polynomial in two or three variables. Some arithmetic properties of such tables will be found, studied and proved. All this was made possible only after successful decryption of the entire class of numeric tables of truncated triangles in the cryptographic system. Results. For example, the arithmetic properties of truncated Pascal’s triangle for finding all prime numbers have been found and presented, and then their formulas have been placed. In addition to elementary addition and subtraction tables, unlimited “comparison” tables of numbers are given and presented for the first time. Conclusions. For computer implementation of the objectives set, the rules of real actions that should exist for tables have been laid down. Only recurrent numeric series should be used for this purpose.

THE COLLATZ CONJECTURE. Order and harmony in the sequence numbers.

I propose a numerical table that demonstrates visually that the sequences formed withCollatz's algorithm always reach 1.

Phase Difference Technique for Material Classification by Using Tuning Fork with Piezoelectric Resonator

A tuning fork with piezoelectric (TFP) is an electro-mechanical device; its principle performance is based on specific frequency and limited voltage. As the principle, the device can result output significantly. This paper presents a material classification using phase difference value that is received from touching between material and modified TFP. For the modified TFP, it is customized by a millimeter needle. The frequency response method on Bode plot is used to observe the modified TFP behavior, and to select a specific frequency. The first-order model with cascade notch filters is employed to identify mathematical model of the TFP, and to verify the model by simulation. Experimentation is considered on 2 touching techniques: a shear-force type and a tapping mode type. There are hard plastic, iron, silicon rubber, vinyl eraser and hydrogel for test material. The results are described by both time series plots of input-output signals and numerical table of shifted phase. The technique can employed to classify the test materials effectively and explicitly.