In an earlier paper published in the Journal of Natural Sciences Research in 2015 on how to balance chemical equations using matrix algebra, Gabriel and Onwuka showed how to reduce the resulting matrix to echelon form...In an earlier paper published in the Journal of Natural Sciences Research in 2015 on how to balance chemical equations using matrix algebra, Gabriel and Onwuka showed how to reduce the resulting matrix to echelon form using elementary row operations. However, they did not show how elementary row operations can be used in reducing the resulting echelon matrix to row reduced echelon form. We show that the solution obtained is actually the nullspace of the matrix. Hence, the solution can be infinitely many. In addition, we show that instead of manually using row operations to reduce the matrix to row reduced echelon form, software environments like octave or Matlab can be used to reduce the matrix directly. In all the examples presented in this paper, we reduced all matrices to row reduced echelon form showing all row operations, which was not clearly stated in the Gabriel and Onwuka paper. Most importantly, with the availability of Mathematical software, we show that we do not need to carry out these row operations by brute force.展开更多
A number of useful techniques associated with two-dimensional correlation spectroscopy(2DCOS)to improve its performance and utility have been developed in the last 30years.Evolution of these 2DCOS techniques,including...A number of useful techniques associated with two-dimensional correlation spectroscopy(2DCOS)to improve its performance and utility have been developed in the last 30years.Evolution of these 2DCOS techniques,including some of the very recent developments,is reviewed with examples.Topics include merged or modified asynchronous 2Dcorrelation spectrum,two-dimensional codistribution spectroscopy(2DCDS),Pareto scaling,and null-space projection treatment of spectral dataset.展开更多
文摘In an earlier paper published in the Journal of Natural Sciences Research in 2015 on how to balance chemical equations using matrix algebra, Gabriel and Onwuka showed how to reduce the resulting matrix to echelon form using elementary row operations. However, they did not show how elementary row operations can be used in reducing the resulting echelon matrix to row reduced echelon form. We show that the solution obtained is actually the nullspace of the matrix. Hence, the solution can be infinitely many. In addition, we show that instead of manually using row operations to reduce the matrix to row reduced echelon form, software environments like octave or Matlab can be used to reduce the matrix directly. In all the examples presented in this paper, we reduced all matrices to row reduced echelon form showing all row operations, which was not clearly stated in the Gabriel and Onwuka paper. Most importantly, with the availability of Mathematical software, we show that we do not need to carry out these row operations by brute force.
文摘A number of useful techniques associated with two-dimensional correlation spectroscopy(2DCOS)to improve its performance and utility have been developed in the last 30years.Evolution of these 2DCOS techniques,including some of the very recent developments,is reviewed with examples.Topics include merged or modified asynchronous 2Dcorrelation spectrum,two-dimensional codistribution spectroscopy(2DCDS),Pareto scaling,and null-space projection treatment of spectral dataset.
