General Mathematics

2004 Submissions

[6] viXra:2004.0654 [pdf] replaced on 2020-07-04 01:41:18

General Formula of Fibonacci Sequence and Lucas Sequence's Reverse Sum

Authors: C. Tungchotiroj
Comments: 4 Pages.

The sum $\sum_{k=1}^{n}{a_kb_{n+1-k}=a_1b_n+a_2b_{n-1}+...+a_nb_1}$, where $n$ are any positive integers, denoted by $R(a_n, b_n)$, are called \textit{Reverse Sum} of $a_n$ and $b_n$. Reverse Sum usually appears in Rearrangement Inequality, but not in normal Algebra. \textit{Fibonacci Sequence} $\{ F_n \}$ and \textit{Lucas Sequence} $\{ L_n \}$ are very similar sequences because they also have recurrence formula, but have $F_0=0$, $F_1=1$ and $L_0=2$, $L_1=0$. Because of that similarity of sequences, we suggest that those sequences can be related as a function of Reverse Sum. In this paper it is shown that $R(F_n, L_n)$ can be written into general form within $\{ F_n \}$ and some various constants.
Category: General Mathematics

[5] viXra:2004.0426 [pdf] submitted on 2020-04-17 13:21:04

Studies in Figured Tours of Knight in Two and Higher Dimensions

Authors: Awani Kumar
Comments: 28 Pages.

Tour of knight is over a millennium year old puzzle but ‘Figured tour’ of knight is a recent field of research. T. R. Dawson, an English chess problemist and the father of Fairy Chess, coined the term in 1940s. The name figured tour is appropriate for any numbered tour in which certain arithmetically related numbers are arranged in a geometrical pattern. Figured tours have been only looked into two-dimensional boards, mostly on 8x8 board. The author has constructed knight tour with square numbers in fiveleaper {3, 4} + {0, 5} path and various other figured tours on 6x6 board and extended it in three and four dimensional space. Construction of figured tours is a mathematical recreation and can also be used in pedagogy of higher mathematics.
Category: General Mathematics

[4] viXra:2004.0300 [pdf] submitted on 2020-04-13 09:09:47

Differential Correction and Arc-Length Continuation Applied to Boundary Value Problems: Examples Based on Snap-Through of Circular Arches and Spherical Shell

Authors: Jun Zhong, Shane D. Ross
Comments: 28 Pages.

Inspired by the application of differential correction to initial-value problems to find periodic orbits in both the autonomous and non-autonomous dynamical systems, in this paper we apply differential correction to boundary-value problems. In the numerical demonstration, the snap-through buckling of arches and shallow spherical shells in structural mechanics are selected as examples. Due to the complicated geometrical nonlinearity in such problems, the limit points and turning points might exist. In this case, the typical Newton-Raphson method commonly used in numerical algorithms will fail to cross such points. In the current study, an arc-length continuation is introduced to enable the current algorithm to capture the complicated load-deflection paths. To show the accuracy and efficiency of differential correction, we will also apply the continuation software package COCO to get the results as a comparison to those from differential correction. The results obtained by the proposed algorithm and COCO agree well with each other, suggesting the validity and robustness of differential correction for boundary-value problems.
Category: General Mathematics

[3] viXra:2004.0065 [pdf] submitted on 2020-04-03 14:40:05

Series for π

Authors: Yuji Masuda
Comments: 1 Page.

This formula shows π.
Category: General Mathematics

[2] viXra:2004.0056 [pdf] submitted on 2020-04-03 02:34:33

A Expression for π

Authors: Yuji Masuda
Comments: 1 Page.

This formula shows π.
Category: General Mathematics

[1] viXra:2004.0018 [pdf] submitted on 2020-04-01 14:54:31

Computation of Glueball Mass Using a Certain Theory

Authors: Kohji Suzuki
Comments: 4 Pages.

We compute the mass of glueball to propose a conjecture.
Category: General Mathematics