Matrix, generally

A sintered friction material is composed of a metal matrix, generally mainly copper, to which a number of other metals such as tin, zinc, lead, and iron are added. Important constituents include graphite and friction-producing components such as siHca, emery, or asbestos. 

Hamilton operator or Hamilton matrix (general, electronic, nuclear)... 

The dimensions of a matrix are given by stating first the number of rows and then the number of columns that it has. Thus, matrix A has six rows and three columns, and is said to be a 6x3 (read six by three ) matrix. Matrix B has one row and three columns and is a 1x3 matrix. Matrix C is a 6x1 matrix. Generally, a matrix M that has r rows and c columns is called an rxc matrix and can be identified as such by the notation... 

Remarkably, when our general ME is applied to either AN or PN in Section 4.4, the resulting dynamically controlled relaxation or decoherence rates obey analogous formulae provided the corresponding density matrix (generalized Bloch) equations are written in the appropriate basis. This underscores the universality of our treatment. It allows us to present a PN treatment that does not describe noise phenomenologically, but rather dynamically, starting from the ubiquitous spin-boson Hamiltonian. 

The terms for the initiators k(j c(/,) and scavengers kSj c(S,) are those with the biggest uncertainty or error. The values of these terms vary depending on the water matrix. Generally, there is not enough kinetic information at hand to use such a complicated model, so that various simplifications have been developed. The following examples illustrate the various approaches that can be taken. 

This technique allows the analysis of only one component or several in the same sample. Standards can be prepared with all components of interest in each standard and the range of composition of the standards should cover the entire range expected in the unknowns. The peak size is then plotted against either absolute amounts of each component or its concentration in the matrix, generally the latter. 

The term foam is defined as a gaseous void surrounded by a much denser continuous matrix, generally a liquid or a solid phase. As a result of the distinct characteristics of the two phases, such cellular materials are able to feature unique properties. Since nature successfully demonstrated their use in manifold examples, such as bones, wood, plant stalks, cork, and sponges, foams have also sparked interest for technical application. Nowadays, a broad range of cellular materials based on metals, ceramics as well as polymers, is readily available, and their structures are as versatile as their applications. 

Starting with a density matrix generalized to different values J and J, the... 

T-matrix approximation developed in Ref. 190 was shown in Refs. 39 and 42. The matrix generalization of this approach was obtained in Refs. 49, 134, and 203. Here we are using its modification proposed in Ref. 125. 

Several additional favorable properties of CBPCs make them an even better candidate for stabilization. The waste form is a dense matrix, generally with very good mechanical properties. Also it is nonleachable, does not degrade over time, is neutral in pH, converts even flammable waste into nonflammable waste forms, performs well within acceptable levels in radiolysis tests, and can incorporate a range of inorganic waste streams (solids, sludge, liquids, and salts). 

Finally, the fluorescent effect that expresses the possibility for an X-ray emitted in the sample to, in turn, excite an atom of the matrix (generally with an atomic number Z - I orZ-2). 

Once an LLS matrix has been identified as poorly conditioned, what are the implications, and what can be done to improve the conditioning if that appears to be desirable The solution to a least-squares problem obtained from a poorly conditioned LLS matrix is, in a word, unreliable. Some components of the solution may be artifacts, and not part of a realistic description of the problem at hand. Reducing the rank of the matrix generally improves its conditioning, and hence the reliability of the solution. However, arriving at an estimate for the actual rank of A and deciding how to reduce the data set such that A becomes full rank are nontrivial problems. 

Since the sample matrix generally gives the same eflFect on the analytical response, as if the analyte would be present in the sample, a positive bias when measured by immunoassay vs. a reference method is often observed [54,55]. Simple mathematical models that parallel the strategy of sample addition for correcting these negative eflFects have been proposed [56,57]. 

A general definition of the Hosoya Z matrix (generalized Hosoya Z matrix) able to represent both acyclic and cyclic graphs is the following [Plavsic et al, 1997] ... 

Hamilton operator or Hamilton matrix (general, electronic, nuclear) Matrix element of a Hamilton operator between Slater determinants Exchange type matrix elements in semi-empirical theory x,y = s,p,d) Summation indices for occupied MOs ... 

For multicomponent systems we may write down the matrix generalization of Eq. 7.3.9 as... 

The only really practical approach is to use the Toor-Stewart-Prober approximation of constant [D]. The starting point for our analysis is the matrix generalization of Eq. 9.1.1... 

For mass transfer in a rigid spherical drop the matrix [F] is given by the n - 1 dimensional matrix generalization of Eq. 9.4.6... 

The analysis of turbulent eddy transport in binary systems given above is generalized here for multicomponent systems. The constitutive relation for j y in multicomponent mixtures taking account of the molecular diffusion and turbulent eddy contributions, is given by the matrix generalization of Eq. 10.3.1... 

Equation 10.4.20 is the matrix generalization of Eq. 10.3.17. To obtain the matrix of low flux mass transfer coefficients [fc] we take the limit as n, goes to zero... 

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