Singular multiples/multiplicities

This set is said to be ill-conditioned because the second equation is almost an exact multiple of the first. The matrix of coefficients is almost singular. 

Although the reactivity function obtained above is of quite general validity, we do not believe that just one such simple, singular function will describe chemical reactivity in a universal manner. Rather, we are presently working with several such functions. Thus, in a similar manner to that described above, additional functions were obtained by LoRA for the reactivity of multiple bonds, and for bonds in charged species. 

The matrix on the right-hand side is singular (its determinant is clearly zero), so that the partition coefficients cannot be independently determined. However, the allotments x of Nd between the two reservoirs can be retrieved by adding the two lines of equation (7.4.7) after multiplication of the first row by xmNd and of the second row by xcNd. The terms involving the partition coefficients cancel out and we get... 

Let s assume the elements ci, a and cz of vector c are the unknowns. Thus, the system is comprised of three equations with three unknowns. Such systems of n equations with n unknowns have exactly one solution if none of the individual equations can be expressed by linear combinations of the remaining ones, i.e. if they are linearly independent. Then, the coefficient matrix A is of full rank and non-singular and its inverse, A1, exists such that right multiplication of equation (2.20) with A 1 allows the determination of the unknowns. 

Matrix inversion is analogous to division. Multiplication of A with its inverse A-1 gives an identity matrix, / (see Figure A.2.6). The inverse is only defined for square matrices that are not singular. A matrix is singular if at least one row (or column) contains equal numbers, or at least one column (or row) is a linear combination of... 

The corresponding quantum mechanical expression of s op in Equation (4.19) is similar except for the quantity Nj, which is replaced by Nfj. However, the physical meaning of some terms are quite different coj represents the frequency corresponding to a transition between two electronic states of the atom separated by an energy Ticoj, and fj is a dimensionless quantity (called the oscillator strength and formally defined in the next chapter, in Section 5.3) related to the quantum probability for this transition, satisfying Jfj fj = l- At this point, it is important to mention that the multiple resonant frequencies coj could be related to multiple valence band to conduction band singularities (transitions), or to transitions due to optical centers. This model does not differentiate between these possible processes it only relates the multiple resonances to different resonance frequencies. 

CNTs can exhibit singular electronic band structures and can show metallic and semiconducting behavior. As a general rule, n.m tubes with n-m being an integer multiple of 3 are metallic, while the remaining tubes are semiconducting. 

Nitrogen and oxygen can be Incorporated Into the backbone such that they are surrounded by different atom types. For example, organic peroxides contain two covalently bonded oxygen atoms that form the peroxide linkage. These molecules are Inherently unstable. Two covalently bonded nitrogen atoms are also similarly unstable. These unstable structures decompose to form smaller unstable molecules that are used to start the polymerization for some types of monomers. Thus, to be incorporated implies that the molecules are found only singularly in the backbone chain. Sulfur and silicon are considered to be chain formers. They can be found in the backbone in multiple units connected covalently to molecules of the same type or with carbon. Complete molecules with a silicon backbone are possible, and molecules with multiple sulfur links incorporated into the system are common, particularly in sulfur-crosslinked rubber. 

It must be emphasized that the computation of at small k is very delicate, and must not be crudely pursued. There is a great deal of structure in the integrand of the multiple wavenumber integrals due to incipient singularities of the bare Coulomb potential and of the repeated energy denominators which are characteristic of perturbation expansions. In fact, the contributions of the individual Feynman graphs had already been calculated analytically in the... 

Boeniger, M.F., Zaebst, D.D., Ludwig, H.R., Crandall, M.S., and Vongrongseman, P., A Field Comparison of Two Passive Organic Vapor Monitors with Charcoal Tubes Under Singular and Multiple Exposure Conditions, Unpublished paper, presented at 1980 American Industrial Hygiene Conference. 

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