Mathematical resolution

Adequate resolution of the components of a mixture in the shortest possible time is nearly always a principal goal. Establishing the optimum conditions by trial and error is inefficient and relies heavily on the expertise of the analyst. The development of computer-controlled HPLC systems has enabled systematic automated optimization techniques, based on statistical experimental design and mathematical resolution functions, to be exploited. The basic choices of column (stationary phase) and detector are made first followed by an investigation of the mobile phase composition and possibly other parameters. This can be done manually but computer-controlled optimization has the advantage of releasing the analyst for other... 

For measurement adjustment, a constrained optimization problem with model equations as constraints is resolved at a fixed interval. In this context, variable classification is applied to reduce the set of constraints, by eliminating the unmeasured variables and the nonredundant measurements. The dimensional reduction of the set of constraints allows an easier and quicker mathematical resolution of the problem. 

More commonly, we are faced with the need for mathematical resolution of components, using their different patterns (or spectra) in the various dimensions. That is, literally, mathematical analysis must supplement the chemical or physical analysis. In this case, we very often initially lack sufficient model information for a rigorous analysis, and a number of methods have evolved to "explore the data", such as principal components and "self-modeling analysis (21), cross correlation (22). Fourier and discrete (Hadamard,. . . ) transforms (23) digital filtering (24), rank annihilation (25), factor analysis (26), and data matrix ratioing (27). 

Gonzalez-Mora JT, Fumero B, Mas M. 1991. Mathematical resolution of mixed in vivo voltammetry signals models, equipment, assesment by simultaneous microdialysis sampling. J neurosci methods 231—244. 

Peak resolution is usually easier if well chosen background parameters are input and if constrained optimization methods are utilised. Misleading results can be obtained if the constraints are too limited and tests with unconstrained optimization are desirable if at all possible. In particular, the possible presence of paracrystalline or intermediate phase peaks must be tested with extreme care in order to avoid ambiguity. It is not sufficient to have a good mathematical resolution alone, all peaks must be significant in crystallographic or structural terms. The incidental measurement of peak-area crystallinity is considered to be of secondary importance to the resolution of overlapping peaks. 

Resolution. The goal of any chromatographic step is to maximize resolution or to minimize zone spreading. Resolution is a function of the position of the maximum elution peak height and the elution peak width (Fig. 2-2). The greater the resolution between two elution peaks, the greater the degree of separation between the two molecules. Mathematically, resolution (R) is two times the distance between two elution peak maxima (d) divided by the sum of the widths of the two elution peaks... 

Fig. 4.1.6. O—H and C—H stretching bands of European beech (Fagus sylvatica) MWL. A 4cm 1 resolution B spectrum after mathematical resolution enhancement using deconvolution technique. (Experimental conditions described in legend to Fig. 4.1.11)...

Static capillary phenomena lead to precisely determined geometrical shapes like sessile menisci, pendent menisci, minimal surfaces, which can be used for the physical determination and measurement of the surface tension or the interfacial tensions between fluids. In addition to the simple forms considered herein, more complex forms (e.g., sessile lenticular drops) can be studied. Mathematical resolution of these shapes is a combination of the (numerical) solution of the highly nonlinear Young-Laplace equation together with an appropriate set of boundary conditions. For practical purposes, only axisymmetric forms are readily amenable to mathematical analysis. 

MEKC-DAD data (54). Studies from Kaniansky and coworkers have focused on using factor analysis, including ITTFA, WFA, and orthogonal projection approach (OPA), for the feasible identification of orotic acid at low concentration level in urine matrices (55, 56). The mathematical resolution of anionic surfactants that cannot be separated electrophoretically has been accomplished by OPA-ALS (57). 

Mathematical resolution of the calibration procedure. Using the coordinates of the i centers obtained in the previous step, we proceed to obtain the matrix of the camera P. The resolution method is divided into two stages In the first stage, we assume the correspondence between a set of 3D points and their 2D counterparts, considering that the optical system has no distortion. Such correspondence exists between the coordinates of the center of grid distortion target dots... 

By changing the outer boundary condition, external resistance can be included in the diffusion model, though this makes the mathematical resolution of the model more difficult. In this case, two parameters must be identified (i) the effective diffusivity (internal resistance) and (ii) the mass transfer coefficient (external resistance). In order to reduce the number of possible solutions, the effective diffusivity identified by neglecting the external resistance can be used as the start value of diffusivity. 

The mathematical resolution of this equation is rather complicated and can be found in [4, 10]. Assuming [Ei] = [EJ = Co at time 0, and neglecting back reaction k+ 3> kJ), reaction half-time ty2 of the reaction is simply ... 

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