Product solution, multidimensional

Product solutions for temperatures In multidimensional systems (a) semi-infinite plate (b) infinite rectangular bar (c) semi-infinite rectangular bar (cf) rectangular parallelepiped (e) semi-infinite cylinder (0 short cylinder. 

We start this chapter with the analysis of lumped systems in which the temperature of a body varies with time but remains uniform throughout at any time. Then we consider the variation of temperature with time as well as position for one-dimensional heat conduction problems such as those associated with a large plane wall, a long cylinder, a sphere, and a semi infinite medium using transient temperature charts and analytical solutions. Finally, we consider transient heat conduction in multidimensional systems by utilizing the product solution. 

That is, the solution for Ihe two-dimensional short cylinder of height a and radiu.s r is equal to the product of the noiidimcusionalized solutions for the oue-dimensional plane wall of thickness a and the long cylinder of radius r , which are the two geouieiiies whose intersection is the short cylinder, as shown in Fig. 4—35. We generalize this as follows the solution for a multidimensional geometry is the product of the solutions of the one-dimensional geometries whose intersection is the multidimensional body. 

In order to pursue heteronuclear multidimensional NMR experiments, a bacterial system for expression of apoLp-III has been developed which allows facile production of 150 - 200 mg/L l N-iabeled apoLp-III or 100 - 125 mg/L 15N/l3c-double labeled apoLp-III. Figure 2, panel A shows the IH- n HSQC spectrum of a 1.0 mM solution of lipid-free, uniformly N-labeled apoLp-III. Panel A also indicates that, although the chemical shift dispersion in the H-dimension is rather small (6.5 ppm to 9.5 ppm), it is generally upfield shifted, consistent with the fact that the protein secondary structure is predominantly a-helix (13). The chemical shifts in the N-dimension are well-dispersed which results in good separation of the overall crosspeaks. However, certain regions in the spectrum are still crowded as shown in Figure 2. 

One-dimensional charts find an important application in the solution of multidimensional problems. It can be shown (see, for example, Section 5.2 of Ret 1) that the dimensionless unsteady temperature of an infinitely long rod of rectangular cross section 21 X 2L may be expressed as the product of the dimensionless temperature of an infinite flat plate of thickness 21 times the dimensionless temperature of an infinite flat plate of thickness 2L,... 

Nuclear magnetic resonance (NMR) spectroscopy evolved into a major technique for the characterization of materials in just about all areas of chemistry. Most researehers and students will be familiar with solution applieations, where chemical shifts and J couplings of spin V2 nuclei can be used for simple spectral analysis of reaction products as well as the complex multidimensional techniques used to determine the structure and conformation of molecules as complex as proteins. A... 

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