Forecasting Carbon Dioxide Emissions for Singapore using Grey Model with Cramer’s Rule

This article analyses and forecasts carbon dioxide () emissions in Singapore for the 2012 to 2016 period. The study analysed the data using grey forecasting model with Cramer’s rule to calculate the best SOGM(2,1) model with the highest accuracy of precision compared to conventional grey forecasting model. According to the forecasted result, the fitted values using SOGM(2,1) model has a higher accuracy precision with better capability in handling information to fit larger scale of uncertain feature compared to other conventional grey forecasting models. This article offers insightful information to policymakers in Singapore to develop better renewable energy instruments to combat the greater issues of global warming and reducing the fossil carbon dioxide emissions into the environment.

A new uncertain linear regression model based on equation deformation

When the observed data are imprecise, the uncertain regression model is more suitable for the linear regression analysis. Least squares estimate can fully consider the given data and minimize the sum of squares of residual error, and can effectively solve the linear regression equation of imprecisely observed data. On the basis of uncertainty theory, this paper presents an equation deformation method for solving unknown parameters in uncertain linear regression equations. We first establish the equation deformation method of one-dimensional linear regression model, and then extend it to the case of multiple linear regression model. We also combine the equation deformation method with Cramer's rule and matrix, and propose the Cramer's rule and matrix elementary transformation method to solve the unknown parameters of the uncertain linear regression equation. Numerical examples show that the equation deformation method can effectively solve the unknown parameters of the uncertain linear regression equation.

A Robust and Fast Image Encryption Scheme Based on a Mixing Technique

This paper introduces a new image encryption scheme using a mixing technique as a way to encrypt one or multiple images of different types and sizes. The mixing model follows a nonlinear mathematical expression based on Cramer’s rule. Two 1D systems already developed in the literature, namely, the May-Gompertz map and the piecewise linear chaotic map, were used in the mixing process as pseudo-random number generators for their good chaotic properties. The image to be encrypted was first of all partitioned into N subimages of the same size. The subimages underwent a block permutation using the May-Gompertz map. This was followed by a pixel-based permutation using the piecewise linear chaotic map. The result of the two previous permutations was divided into 4 subimages, which were then mixed using pseudo-random matrices generated from the two maps mentioned above. Tests carried out on the cryptosystem designed proved that it was fast due to the 1D maps used, robust in terms of noise and data loss, exhibited a large key space, and resisted all common attacks. A very interesting feature of the proposed cryptosystem is that it works well for simultaneous multiple-image encryption.

Optimized Cramer’s Rule in WZ Factorization and Applications

In this paper, W Z factorization is optimized with a proposed Cramer’s rule and compared with classical Cramer’s rule to solve the linear systems of the factorization technique. The matrix norms and performance time of WZ factorization together with LU factorization are analyzed using sparse matrices on MATLAB via AMD and Intel processor to deduce that the optimized Cramer’s rule in the factorization algorithm yields accurate results than LU factorization and conventional W Z factorization. In all, the matrix group and Schur complement for every Zsystem (2×2 block triangular matrices from Z-matrix) are established.

Cramer’s rule over quaternions and split quaternions: A unified algebraic approach in quaternionic and split quaternionic mechanics

This paper aims to present, in a unified manner, Cramer’s rule which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies Cramer’s rule for the system of v-quaternionic linear equations by means of a complex matrix representation of v-quaternion matrices, and gives an algebraic technique for solving the system of v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for Cramer’s rule in quaternionic and split quaternionic mechanics.