Theorem of superposition

Hermann von Helmholtz was a student of Du Bois-Reymond. He measured the conduction velocity of a nerve cell axon around 1850. He formulated the very basic theorems of superposition and reciprocity, and also some very important laws of... 

The theorem of superposition When the state of stress in a body is other than a simple, normal ... 

Then, considering the effect of each stress component acting separately and using the theorem of superposition, gives for the linear strains when both stress components act simultaneously ... 

This gives the basic theorem in the method of superposition of configurations ... 

Thus, the use of wave function in the form (23.67) and operators in the form (23.62)—(23.66) makes it possible to separate the dependence of multi-configuration matrix elements on the total number of electrons using the Wigner-Eckart theorem, and to regard this form of superposition-of-configuration approximation itself as a one-configuration approximation in the space of total quasispin angular momentum. 

According to the superposition theorem of system theory for linear responses, this response to a step-function in the current can be employed to deduce the impedance behavior. As regards a qualitative discussion, one can adopt the above description by just replacing short/long times by high/small frequencies. Quantitatively the impedance is given by a Laplace transformation of Eq. (64) (or equivalently by applying Kirchhoff s laws to the equivalent circuit (Eq. (63))) with the result... 

Figure 4.32 illustrates macroscopic anisotropy in a simplified tissue model. In living tissue, conductivity may be 10 times larger in one direction than another. At low amplitude levels the tissue is still linear, and the principle of superposition and the reciprocity theorem are still valid. However, Ohm s law for volume conductors, J = oE, is not necessarily valid even if it still is linear the current density direction will not coincide with the E-field direction if the anisotropic structures are sufficiently small. 

To see the value of this alternative view of superposition, we shall derive the conserved charge corresponding to this symmetry implied by Noether s theorem. Using the fact that x / and ( ) both obey the wave equation, the Lagrangian density (Equation 4.51) transforms under Equation 4.75 as... 

A theorem which, at first sight, does not seem to be very closely related to Polya s Theorem, but which in fact has much affinity with it, is the superposition theorem that appeared in my doctoral thesis [ReaR58] and later in [ReaR59,60]. The general problem to which it applies is the following. Consider an ordered set of k permutation groups of degree , say G. G. . and the set of all A -ads... 

The superposition theorem then gives a method of determining the number of equivalence classes under this relation of similarity. 

This trick of manipulating two or more families of variables simultaneously can be used also with applications of the superposition theorem. This idea seems to have been discovered independently by several workers for an example see [MulI69], where it is used to enumerate a class of regular digraphs. 

Powerful though Polya s Theorem undoubtedly is, it is not difficult to formulate problems which, though superficially very similar to the general Polya-type problem, cannot be solved by use of the theorem. We have already seen how some problems of this kind led to generalizations of Polya s Theorem, such as de Bruijn s Theorem, the power group enumeration theorem, and the superposition theorem, but even these theorems have their limitations. For some problems not amenable to solution by any of these results, it is sufficient to fall back on the result from which they all stem, namely Burnside s Lemma. 

Equation (23) implies that the current density is uniformly distributed at all times. In reality, when the entire electrode has reached the limiting condition, the distribution of current is not uniform this distribution will be determined by the relative thickness of the developing concentration boundary layer along the electrode. To apply the superposition theorem to mass transfer at electrodes with a nonuniform limiting-current distribution, the local current density throughout the approach to the limiting current should be known. 

Collective modes can be viewed as superpositions of Iph configurations. It is convenient to define this relation by using the Thouless theorem which establishes the connection between two arbitrary Slater determinants [25]. Then, the perturbed many-body wave function reads... 

Superposition may be invoked to determine the behavior of the theorem when the functions are subjected to changes in the sign of real or imaginary, odd or even, components. 

These expressions are formally exact and the first equality in Eq. (123) comes from Euler s theorem stating that the AT potential u3(rn, r23) is a homogeneous function of order -9 of the variables r12, r13, and r23. Note that Eq. (123) is very convenient to realize the thermodynamic consistency of the integral equation, which is based on the equality between both expressions of the isothermal compressibility stemmed, respectively, from the virial pressure, It = 2 (dp/dE).,., and from the long-wavelength limit S 0) of the structure factor, %T = p[.S (0)/p]. The integral in Eq. (123) explicitly contains the tripledipole interaction and the triplet correlation function g (r12, r13, r23) that is unknown and, according to Kirkwood [86], has to be approximated by the superposition approximation, with the result... 

Once this discussion of the space-inversion operator in the context of optically active isomers is accepted, it follows that a molecular interpretation of the optical activity equation will not be a trivial matter. This is because a molecule is conventionally defined as a dynamical system composed of a particular, finite number of electrons and nuclei it can therefore be associated with a Hamiltonian operator containing a finite number (3 M) of degrees of freedom (variables) (Sect. 2), and for such operators one has a theorem that says the Hamiltonian acts on a single, coherent Hilbert space > = 3 (9t3X)51). In more physical terms this means that all the possible excitations of the molecule can be described in . In principle therefore any superposition of states in the molecular Hilbert space is physically realizable in particular it would be legitimate to write the eigenfunctions of the usual molecular Hamiltonian, Eq. (2.14)1 3 in the form of Eq. (4.14) with suitable coefficients (C , = 0. Moreover any unitary transformation of the eigen-... 

---