Logarithmic form

Abstract: We obtain the operator generalization of the theorem on the logarithmic residue for meromorphic operator-functions. The proof of the generalization is based on a theorem concerning a special factorization of a meromorphic operator-function at a point. This theorem also allows us to generalize, to the case of meromorphic operator-functions, the formula of M. V. Keldys for the principal part of the resolvent as well as several other theorems.A definition is given for the multiplicity of a pole for a meromorphic operator-function. The basic properties of the multiplicity of a pole are proved, and also a generalization of the Rouche theorem.Bibliography: 16 items.

Abstract: posed of two intersecting segments, Smith [1977] and Smith and McLean [1977] demonstrate that multiple roughness scales can generate velocity profiles with more than two segments. Thus when form drag is important the constantstress assumption will not be valid. In an earlier paper [Caldwell and Chriss, 1979], we demonstrated the existence of the viscous sublayer in the bottom boundary layer and found a logarithmic layer above it. In examining additional data from the experiment, segmented profiles were found in the logarithmic region (Figures 1 and 2), as expected if form drag influences the flow. When the original data set was reanalyzed, using thinner averaging intervals in the upper portion of the profile and also incorporating both upward and downward traverses, it too shows two distinct logarithmic slopes. (In the original study, only downward traverses were used.) Although deviation from a single logarithmic form is not necessarily large, the slope of the logarithmic regression is significantly different in the two regions, implying that the turbulent stress above is significantly larger than it is nearer the bed.

Abstract: The article is a response to a recent opinion piece that log concentration values should not be applied in analytical chemistry. An essential aim in the development of analytical chemistry methods is to obtain more sensitive and accurate detection values. For the application of chemical analysis methods, the obtained experiment data need to fit with the mathematical functions in the first place. As influenced by different detection principles and analytical methods, data can be displayed in a coordinate system with two linear axes for linear function fitting, or the data can first be taken through a logarithmic transformation and then for function fitting. Using raw data or data after logarithmic transformation primarily depends on analytical principles, without special rules of data formats. For example, ultraviolet-visible spectrophotometric data are more suitable for direct linear fitting. However, enzyme-catalyzed reaction or electrochemical data in logarithmic form are more appropriate for function fitting. This transformation of data form will not affect the soundness of fit statistics; rather, it simplifies the complexity of function analysis and calculation, which are the essence of analytical chemistry. In this brief article, we provide justification and legitimacy of the application of logarithmic processing in various fields of quantitative analytical chemistry.