Recall that if m : M [right arrow] M is a mapping and [OMEGA] [subset] [PHI](M) a submodule, then m* [OMEGA] [subset] [PHI] (M) denotes the submodule with generators m* [omega] ([omega] [member of] [OMEGA]) in accordance with the common practice in the

algebraical module theory.

On the real and imaginary roots of

algebraical equations.

Dot product of two Cartesian vectors are multiple of the corresponding x, y, z components and their

algebraical sum.

By deforming the integration contours in (35) and (36) up to the banks of the cut [C.sub.1] which is the line of the steepest descent and calculating the short-wave asymptotic behavior of the integral, employing the saddle point method and the etalon integral (37), the system of integral Equations (35), (36) is reduced to a system of

algebraical linear equations:

At the end of the chapter on the Arab revolt in Oriental Assembly (also contained in The Seven Pillars of Wisdom), Lawrence helpfully sums up his view of insurgent warfare in 50 words: "Granted mobility, security (in the form of denying targets to the enemy), time, and doctrine (the idea to convert every subject to friendliness), victory will rest with the insurgents, for the

algebraical factors are in the end decisive, and against them perfections of means and spirit struggle quite in vain." (14)

* Although the results discussed are also valid (at the

algebraical level) in case T = 1, this case cannot be tested empirically, because CML-estimates in the PCM do not exist if the test is composed of one partial credit item.

The differential and

algebraical equations of the motion of this body are:

Mention should be made that, based on these results, the stretches do not seem to be tensors in their

algebraical nature, as usually taken for granted.

Before attempting the line concept, the concept of slope must be introduced in a simple way, and with the clear

algebraical movements that the professor introduces in the classroom the concept of equation y=mx+b can be reached.

In general, three major approaches have been used (Yeh and Tripathi, 1989) which include: (1) Coupling existing geochemical or microbiological code to solute transport codes; (2) Direct Substitution (DSA) of the nonlinear chemical reactions into the transport equations leading to a set of nonlinear Partial Differential Equations (PDE's); and (3) Operator Splitting and Sequential Iteration Approach (SIA) in which linear transport PDE's and nonlinear

Algebraical Equations (AE's) are solved sequentially in an iterative manner (multiple-step method or SIA) or without iterations (two-step method or operator splitting).

By this method of antagonism we steady our minds, not so as to reach their object, but to point them in the right direction; as in an

algebraical process we might add and subtract in series, approximating little by little, by saying and unsaying, to a positive result.

That is a mathematical truth--a proposition of analytic mechanics; and anybody can see without any

algebraical apparatus that mechanical law out of like antecedents can only produce like consequents.

Algebraical symbols are used when you do not know what you are talking about.

answers to no real thing, but merely serves as an

algebraical symbol for comparing the values of products with one another" [Bellamy, 1887, p.