The Functionality Matrix

Form Designation Functional Form Equation Form 

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One method used to summarize the required devices and show the function performed by each device is with a function matrix. Figure 14-4 is a completed function matrix chart for the simple process flow diagram shown in Figure 14-5. The function matrix is from RP I4C and is called a SAFE chart. Each component is listed in the left hand column with an identification number and description. Under Device I.D., each of the devices listed in the SAC is listed. If the device is not present, the appropriate SAC reference number is listed. If the SAC rationale requires that another device be present on another component, that device is listed under Alternate Device, if applicable. 

In addition to the insoluble polymers described above, soluble polymers, such as non-cross-linked PS and PEG have proven useful for synthetic applications. However, since synthesis on soluble supports is more difficult to automate, these polymers are not used as extensively as insoluble beads. Soluble polymers offer most of the advantages of both homogeneous-phase chemistry (lack of diffusion phenomena and easy monitoring) and solid-phase techniques (use of excess reagents and ease of isolation and purification of products). Separation of the functionalized matrix is achieved by either precipitation (solvent or heat), membrane filtration, or size-exclusion chromatography [98,99]. 

Moss, M.L. (1962) The Functional Matrix. Vistas in Orthodontics. Editors B. Kraus and R. Reidel 85-98, Lea and Febiger, Phila... 

Moss, M.L. (1969) A theoretical analysis of the functional matrix. Acta Biotheoretica 18 195-202... 

Moss, M.L. (1997a) The functional matrix hypothesis revisited. l.The role of mechano-transduction. American Journal of Orthodontal and Dentofacial Orthopedics 112 8-11... 

If, in addition, one can show that R = r = n, a unique solution of Elq. (B.47) exists for the set of Lagrangian multipliers Aj. However, whether the rank of the functional matrix p equals the number of constraints depends on the specific problem and needs to be investigated separately for each case under consideration. 

Chemistry on soluble polymer matrices has recently emerged as a viable alternative to solid-phase organic synthesis (SPOS) involving insoluble cross-linked polymer supports. Separation of the functionalized matrix is achieved by solvent or heat precipitation, membrane filtration, or size-exclusion chromatography. Suitable soluble polymers for liquid phase synthesis should be crystalline at room temperature, with functional groups on terminal ends or side chains, but must not be not cross-linked they are therefore soluble in several organic solvents. 

G[x(t), v(t), t] the functional matrix (describing the control system u(t) the control variables veetor (status of the control system)... 

To develop a simultaneous Newtonian procedure for multiconponent distillation, Naphtali and Sandholm 119711 first wrote the N(2C + 1) equations and variables consisting of conponent mass balances [essentially Eq. 16-131 but without substitution of the equilibrium K values for y ], the energy balances [essentially Eq. 16-2811. and equilibrium relationships including Murphree vapor efficiencies in functional matrix form. Their procedure is illustrated for the energy balance. Essentially, the functional matrix form for Eq. 16-281 is... 

The example functional matrix of Figure 2.24a demonstrates the effects of the functional forms of equations. The functionality matrix (Figure 2.24a) is a representation of the following equations ... 

The solution sequence represented by the functionality matrix of Figure 2.24f (x3 is the design variable) is equal in difficulty structurally to... 

The results of running the equation ordering algorithm is the functionality matrix of Figure 2.31. There are two hierarchies which indicate that there are two iterative variables needed to solve this problem or that some of the equations must be solved simultaneously. The step equations will be located at rows 5 and 3 of the rearranged functionality matrix. 

Perform a structural analysis on the functionality matrix in Figure 2.36. 

Where the above application required calixarenes that could bind essentially all proteins with equal affinity, there is also interest in the use of substituted calixarenes as specific binding agents for certain protein partners. An unbiased approach to this problem starts with a specific calixarene and interrogates the whole proteome for specific binding partners. Tetraacetoamidocalixarene (18) was linked by its lower rim to solid support. The functionalized matrix was used to pull down partners from cell lysate. In order to identify positive targets, the pulled down fraction was processed and analyzed by conventional LC-MS/MS-driven proteomics (Fig. 23.24). One specific partner, protein disulfide isomerase (PDl) was then subject to fiuther confirmation and independent experiments that proved that the calixarene could bind and modulate its biochemical activity. This novel protein-fishing technique could easily be expanded to differently functionalized calixarenes [51]. 

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