Expanded coefficient

Here TV" is the total number of energy levels , and the expanding coefficients are... 

Besides these stochastic interpretations, deterministic interpretations are presently developed GRAY [12], KUMPINSKY and EPSTEIN [13], propose systemic approaches, commonly used in chemical engineering several ideal reactors are coupled by conservative flows with expandable coefficients, so that by-passes or dead zones may be taken into account. NICOLIS and FRISCH 14] use a quasi-Semenov equation in the limit of large diffusion coefficients and obtain a renormalization of k , DEWEL et al. [15] use a phenomenological theory of turbulent mixing to study surface effects produced by the feed of the reactor. 

It is worthwhile, albeit tedious, to work out the condition that must satisfied in order for equation (A1.1.117) to hold true. Expanding the trial fiinction according to equation (A1.1.113). assuming that the basis frmctions and expansion coefficients are real and making use of the teclmiqiie of implicit differentiation, one finds... 

The electrostatic potential generated by a molecule A at a distant point B can be expanded m inverse powers of the distance r between B and the centre of mass (CM) of A. This series is called the multipole expansion because the coefficients can be expressed in temis of the multipole moments of the molecule. With this expansion in hand, it is... 

The strategy for representing this differential equation geometrically is to expand both H and p in tenns of the tln-ee Pauli spin matrices, 02 and and then view the coefficients of these matrices as time-dependent vectors in three-dimensional space. We begin by writing die the two-level system Hamiltonian in the following general fomi. 

Here the coefficients G2, G, and so on, are frinctions ofp and T, presumably expandable in Taylor series around p p and T- T. However, it is frequently overlooked that the derivation is accompanied by the connnent that since. . . the second-order transition point must be some singular point of tlie themiodynamic potential, there is every reason to suppose that such an expansion camiot be carried out up to temis of arbitrary order , but that tliere are grounds to suppose that its singularity is of higher order than that of the temis of the expansion used . The theory developed below was based on this assumption. 

Given a set of A -electron space- and spin-synnnetty-adapted configuration state fiinctions in tenns of which is to be expanded as T = S. Cj two primary questions arise (1) how to detemiine the 9 coefficients and the energy E and (2) how to find the best spin orbitals ( ). ] Let us first consider the 1 where a single configuration is used so only the question of detemiining the spin orbitals exists. 

The density is computed as p(r) = 2. n i ). (/ )p. Often, p(r) is expanded in an AO basis, which need not be the same as the basis used for the and the expansion coefficients of p are computed in tenns of those of the It is also connnon to use an AO basis to expand p (r) which, together with p, is needed to evaluate the exchange-correlation fiinctionaTs contribution toCg. 

Each logarithm in the last temi can now be expanded and the (—n)th Fourier coefficient arising fi om each logarithm is — jn) zk-Y- To this must be added the n = 0 Fourier coefficient coming from the first, f-independent term and that arising from the expansion of second term as a periodic function, namely. 

After substitution of the leading terms of the expanded variables into the model equations and equating coefficients of equal powers of e from their sides, they are divided by common factors to obtain the following set ... 

Thermal Stresses. When the wak of a cylindrical pressure vessel is subjected to a temperature gradient, every part expands in accordance with the thermal coefficient of linear expansion of the steel. Those parts of the cylinder at a lower temperature resist the expansion of those parts at a higher temperature, so setting up thermal stresses. To estimate the transient thermal stresses which arise during start-up or shutdown of continuous processes or as a result of process intermptions, it is necessary to know the temperature across the wak thickness as a function of radius and time. Techniques for evaluating transient thermal stresses are available (59) but here only steady-state thermal stresses are considered. The steady-state thermal stresses in the radial, tangential, and axial directions at a point sufficiently far away from the ends of the cylinder for there to be no end effects are as fokows ... 

In some Hquid crystal phases with the positional order just described, there is additional positional order in the two directions parallel to the planes. A snapshot of the molecules at any one time reveals that the molecular centers have a higher density around points which form a two-dimensional lattice, and that these positions are the same from layer to layer. The symmetry of this lattice can be either triangular or rectangular, and again a positional distribution function, can be defined. This function can be expanded in a two-dimensional Fourier series, with the coefficients in front of the two... 

The unknown solution is expanded in a series of known functions [bi x) with unknown coefficients (<7 ]. 

Expanded joints (Fig. 10-138) are confined to the smaller pipe sizes of ductile metals. A smooth finish is required on the outside of the pipe and on the faces of the ridges inside the bore. Pipe and bore must have the same coefficient of thermal expansion. Furthermore, it is essential that the pipe metal have a lower yield point than the metal... 

Any bi-niclal combination, having large differences in their coefficients of linear expansion, such as a bimetal of brass and steel is used for sueh applications. One end of a strip is fixed and the other is left free for natural movement. When heated, brass expands more than the steel and bends towards the steel as shown, giving the desired movement to actuate a tripping lever. 

Hygroscopic (moisture) effects arise for polymer materials such as some epoxies that absorb moisture chemically after curing and therefore expand. These effects are directly analogous to thermal effects and are characterized by coefficients of moisture expansion and p2 in principal material coordinates in direct analogy to a.( and 02 for coefficients of thermal expansion. All calculations for thermal effects with the a can be replaced by or supplemented with analogous terms for moisture expansion. 

---