Dimensional matrix

An interesting development of the PHB technique leads to four-dimensional data storage. By variation of an electric field appHed to the sample the spectral profile of the absorption holes can specifically be altered. This adds two more dimensions to the geometrically two-dimensional matrix frequency of laser light and electrical field strength (174). 

Additional hydrolysis to promote polymerisation and cross-linking leading to a three-dimensional matrix and gel formation. 

An appropriate set of iadependent reference dimensions may be chosen so that the dimensions of each of the variables iavolved ia a physical phenomenon can be expressed ia terms of these reference dimensions. In order to utilize the algebraic approach to dimensional analysis, it is convenient to display the dimensions of the variables by a matrix. The matrix is referred to as the dimensional matrix of the variables and is denoted by the symbol D. Each column of D represents a variable under consideration, and each tow of D represents a reference dimension. The /th tow andyth column element of D denotes the exponent of the reference dimension corresponding to the /th tow of D ia the dimensional formula of the variable corresponding to theyth column. As an iEustration, consider Newton s law of motion, which relates force E, mass Af, and acceleration by (eq. 2) ... 

The dimensional matrix associated with Newton s law of motion is obtained as (eq. 3)... 

As indicated earlier, the vaUdity of the method of dimensional analysis is based on the premise that any equation that correcdy describes a physical phenomenon must be dimensionally homogeneous. An equation is said to be dimensionally homogeneous if each term has the same exponents of dimensions. Such an equation is of course independent of the systems of units employed provided the units are compatible with the dimensional system of the equation. It is convenient to represent the exponents of dimensions of a variable by a column vector called dimensional vector represented by the column corresponding to the variable in the dimensional matrix. In equation 3, the dimensional vector of force F is [1,1, —2] where the prime denotes the matrix transpose. 

Theorem 1. The number of products in a complete set of B-numbers associated with a physical phenomenon is equal to n — r, where n is the number of variables that are involved in the phenomenon and ris the rank of the associated dimensional matrix. 

To show the equivalence of Theorems 1 and 3, it is only necessary to demonstrate that the maximum number of the variables that will not form a dimensionless product is equal to the rank of the dimensional matrix D. 

In terms of linear vector space, Buckingham s theorem (Theorem 2) simply states that the null space of the dimensional matrix has a fixed dimension, and Van Driest s rule (Theorem 3) then specifies the nullity of the dimensional matrix. The problem of finding a complete set of B-numbers is equivalent to that of computing a fundamental system of solutions of equation 13 called a complete set of B-vectors. For simplicity, the matrix formed by a complete set of B-vectors will be called a complete B-matrix. It can also be demonstrated that the choice of reference dimensions does not affect the B-numbers (22). 

In example 1, there are four variables that are involved in the pendulum problem. The associated dimensional matrix Dis given in equation 15. Since... 

Suppose now that force length and time t ate chosen as the reference dimensions. From Table 1 the new dimensional matrix D, becomes (eq. 19)... 

The matrix D , that will tiansfomi D, to D is the dimensional matrix of the variables force, length, and time with respect to the reference dimensions and t. Again from Table 1 equation 20 is obtained. 

Once the dimensional matrix has been set up and the number of products in a complete set of B-numbers determined, a complete set of B-vectors must be computed. In the following, a systematic procedure for this purpose is presented. 

Let Dhe the dimensional matrix of order mhy n associated with a set of variables of a physical phenomenon, where m is the number of chosen reference dimensions and n the number of variables of the set. Without loss of generaUty, it may be assumed that n > m. Consider the augmented matrix (eq. 21) ... 

Pxampk 2. A smooth spherical body of projected area Al moves through a fluid of density p and viscosity p with speed O. The total drag 8 encountered by the sphere is to be determined. Clearly, the total drag 8 is a function of O, Al, p, and p. As before, mass length /, and time t are chosen as the reference dimensions. From Table 1 the dimensional matrix is (eq. 23) ... 

Siace the columns of any complete B-matrix are a basis for the null space of the dimensional matrix, it follows that any two complete B-matrices are related by a nonsingular transformation. In other words, a complete B-matrix itself contains enough information as to which linear combiaations should be formed to obtain the optimized ones. Based on this observation, an efficient algorithm for the generation of an optimized complete B-matrix has been presented (22). No attempt is made here to demonstrate the algorithm. Instead, an example is being used to illustrate the results. 

Example 4. For a given lattice, a relationship is to be found between the lattice resistivity and temperature usiag the foUowiag variables mean free path F, the mass of electron Af, particle density A/, charge Planck s constant Boltzmann constant temperature 9, velocity and resistivity p. Suppose that length /, mass m time /, charge and temperature T are chosen as the reference dimensions. The dimensional matrix D of the variables is given by (eq. 55) ... 

Corollary.—Any four-dimensional matrix can be written as a linear combination of these sixteen linearly independent matrices. 

A hydrogel is formed by a water-soluble polymer that has been lightly crosslinked. Hydrogels swell as they absorb water but they do not dissolve. The volume expansion is limited by the degree of crosslinking. The minimum number of crosslinks needed to form a three-dimensional matrix is approximately 1.5 crosslinks per chain, and this yields the maximum expansion possible without separation of the chains into a true solution. Thus, a hydrogel may be more than 95% water and, in that sense, has much in common with living soft tissues. 

Setton and Chilkoti applied ELPs as a three-dimensional matrix to entrap chondrocytes. In their study, ELP[VsG3A2-90] with a transition temperature of 35°C at 50 mg/mL in PBS was used. This biopolymer can be used to generate a suspension with cells, which upon injection into a defect site will form a scaffold. They showed that in vitro the resulting ELP gel supported the viability of chondrocytes and the synthesis and accumulation of cartilage-specific extracellular matrix material. This suggested that ELPs indeed could be used for in situ formation... 

Zeolites are prepared by the linking of basic structural units around a template molecule. The structural units are typically based on oxides of silicon and aluminium, and the templates are usually individual small molecules. Under the right conditions, the silicon and aluminium oxide precursors will link up around the template to form a crystalline three-dimensional matrix containing the template molecules. The template... 

The sensitivity matrix, G(t), is a (3xl)-dimensional matrix with elements ... 

Interaction matrix A two-dimensional matrix listing all components of interest on the x and y axes and recording the consequences of mixing of these components for each combination of the components. It is useful for identifying chemical reaction hazards and incompatibilities. 

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