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# Inquiry Driven Systems • Fields Of Inquiry

Author: Jon Awbrey

 The field denotes this body, and wise men call one who knows it the field-knower. — Bhagavad Gita, 13.1

## Introduction : Review and Transition

 Know me as the field-knower in all fields — what I deem to be knowledge is knowledge of the field and its knower. — Bhagavad Gita, 13.2

In this exposition the character of a logical expansion or analytic form is seen to correspond to a particular perspective on a universe of discourse, a specialized way of viewing its propositions and organizing their interpretations into comprehensible forms.

In this essay we study the use of analytic forms both in the ordinary expansions and the differential analysis of propositions. The process of logical expansion is formalized in greater detail and, to compensate for the extra level of abstraction, the resulting analytic forms are motivated on intuitive lines as variant perspectives or alternate ways of viewing the situations represented in propositions. By the end of this ascent we hope the reader finds it pleasing to contemplate the whole panorama of differential operations and group actions on a proposition as just so many facets of its differential enlargement which arise from its projection onto complementary perspectives.

### Proposition Fields

 Hear from me in summary what the field is in its character and changes, and of the field-knower's power. — Bhagavad Gita, 13.3

This section introduces the class of mathematical objects known as proposition fields. These are structures which arise in a natural way from the analysis of information systems, especially when it comes to systems with “reflective” intelligence, those which maintain and process components of information about their own states. A system of this kind has its dynamics best understood only if we interpret some of its dimensions of variation as reflecting information about other components of state, and further, only if we assume that the system itself “interprets” these variables as being informative in just this way.

A notion of intelligence has just been made to depend on a notion of interpretation. Both issues are to be side-stepped here, but we leave this note as a pointer to future work. To make these ideas explicit, we ought to say what it means for a system to be an interpreter of some qualities of its state as signs for other qualities of its state. But understanding how this is possible is tantamount to a major objective of this whole study, and can only be developed over the course of our investigation. It may not seem like good mathematical practice to let the clearing up of definitions fall to the end, but that is how it sometimes has to be.

A proposition field is a mapping from points to propositions, that is, it is a function having the general form ${\displaystyle U\to (V\to \mathbb {B} ).\!}$ We may visualize the proposition field as “tagging” each point ${\displaystyle x\!}$ in one universe of discourse ${\displaystyle U\!}$ with a determinate proposition ${\displaystyle f_{x}:V\to \mathbb {B} ~\!}$ in another universe of discourse ${\displaystyle V.\!}$

Under the general type of a proposition field there are numerous special cases which frequently arise. Common species of proposition fields are listed in Table 1, which displays next to the informal name of each an indication how its type is specialized.

 ${\displaystyle {\text{Name}}\!}$ ${\displaystyle {\text{Type}}\!}$ ${\displaystyle {\text{Field}}\!}$ ${\displaystyle U\to (V\to \mathbb {B} )\!}$ ${\displaystyle {\text{Fold}}\!}$ ${\displaystyle U\to (U\to \mathbb {B} )\!}$ ${\displaystyle {\text{File}}\!}$ ${\displaystyle U\to (U^{\prime }\to \mathbb {B} )\!}$ ${\displaystyle {\text{Flow}}\!}$ ${\displaystyle U\to (\mathrm {d} U\to \mathbb {B} )\!}$ ${\displaystyle {\text{Fray}}\!}$ ${\displaystyle \mathrm {d} U\to (U\to \mathbb {B} )\!}$

In the midst of a particular application we often find ourselves taking a definite proposition ${\displaystyle F\!}$ of a certain complexity and deriving from it a proposition field ${\displaystyle {\boldsymbol {\iota }}F.\!}$ The proposition field ${\displaystyle {\boldsymbol {\iota }}F\!}$ may be viewed as setting forth one of the many ways of comprehending ${\displaystyle F\!}$ as an organized collection of simpler propositions, namely, by associating a unique proposition on a simpler space with each point of another simpler space.

The proposition ${\displaystyle F,\!}$ in turn, often arises as a tacit extension ${\displaystyle {\boldsymbol {\varepsilon }}f\!}$ or a differential enlargement ${\displaystyle \mathrm {E} f\!}$ of a prior proposition ${\displaystyle f.\!}$ Table 2 exhibits typical settings in which proposition fields are developed and applied, taking into consideration the kinds of fields noted above. In each case, composing the whole sequence of operations found in a given row of the Table, we define a transformation ${\displaystyle T,\!}$ parameterized by the subscripted argument of the last evaluation step, which takes us from propositions in one universe of discourse to propositions in that same or other related universe.

 ${\displaystyle {\text{Origin}}~f\!}$ ${\displaystyle \to ~\!}$ ${\displaystyle {\text{Extension}}~F~\!}$ ${\displaystyle \to ~\!}$ ${\displaystyle {\text{Factorization}}~\Gamma \!}$ ${\displaystyle \to ~\!}$ ${\displaystyle {\text{Evaluation}}~g\!}$ ${\displaystyle f:U\to \mathbb {B} \!}$ ${\displaystyle {\boldsymbol {\varepsilon }}\!}$ ${\displaystyle {\boldsymbol {\varepsilon }}f:U\times V\to \mathbb {B} \!}$ ${\displaystyle {\boldsymbol {\iota }}_{1}\!}$ ${\displaystyle \Phi f:U\to (V\to \mathbb {B} )\!}$ ${\displaystyle {\boldsymbol {\varphi }}_{u}\!}$ ${\displaystyle T_{u}f:V\to \mathbb {B} \!}$ ${\displaystyle f:U\to \mathbb {B} \!}$ ${\displaystyle {\boldsymbol {\varepsilon }}\!}$ ${\displaystyle {\boldsymbol {\varepsilon }}f:U\times U\to \mathbb {B} \!}$ ${\displaystyle {\boldsymbol {\iota }}_{1}\!}$ ${\displaystyle \Phi f:U\to (U\to \mathbb {B} )\!}$ ${\displaystyle {\boldsymbol {\varphi }}_{u}\!}$ ${\displaystyle T_{u}f:U\to \mathbb {B} \!}$ ${\displaystyle f:U\to \mathbb {B} \!}$ ${\displaystyle {\boldsymbol {\varepsilon }}\!}$ ${\displaystyle {\boldsymbol {\varepsilon }}f:U\times U^{\prime }\to \mathbb {B} \!}$ ${\displaystyle {\boldsymbol {\iota }}_{1}\!}$ ${\displaystyle \Phi f:U\to (U^{\prime }\to \mathbb {B} )\!}$ ${\displaystyle {\boldsymbol {\varphi }}_{u}\!}$ ${\displaystyle T_{u}f:U^{\prime }\to \mathbb {B} \!}$ ${\displaystyle f:U\to \mathbb {B} \!}$ ${\displaystyle \mathrm {E} \!}$ ${\displaystyle \mathrm {E} f:U\times \mathrm {d} U\to \mathbb {B} \!}$ ${\displaystyle {\boldsymbol {\iota }}_{1}\!}$ ${\displaystyle \Phi f:U\to (\mathrm {d} U\to \mathbb {B} )\!}$ ${\displaystyle {\boldsymbol {\varphi }}_{u}\!}$ ${\displaystyle T_{u}f:\mathrm {d} U\to \mathbb {B} \!}$ ${\displaystyle f:U\to \mathbb {B} \!}$ ${\displaystyle \mathrm {E} \!}$ ${\displaystyle \mathrm {E} f:U\times \mathrm {d} U\to \mathbb {B} \!}$ ${\displaystyle {\boldsymbol {\iota }}_{2}\!}$ ${\displaystyle \Psi f:\mathrm {d} U\to (U\to \mathbb {B} )\!}$ ${\displaystyle {\boldsymbol {\varphi }}_{v}\!}$ ${\displaystyle T_{v}f:U\to \mathbb {B} \!}$

### Analytic Expansions

 Ancient seers have sung of this in many ways, with varied meters and with aphorisms on the infinite spirit laced with logical arguments. — Bhagavad Gita, 13.4

In “Tools and Views” we considered the analytic expansions of propositions with respect to various degrees of interpretation of their variables. The significance of these expansions for logical understanding is that they factor the process of interpretation between two stages, in application giving two moments to the clarification of a problematic proposition. In type, an expansion corresponds to an isomorphism,

 ${\displaystyle {\begin{matrix}(\mathbb {B} ^{n}\to \mathbb {B} )&\cong &(\mathbb {B} ^{j}\times \mathbb {B} ^{n-j}\to \mathbb {B} ))&\cong &(\mathbb {B} ^{j}\to (\mathbb {B} ^{n-j}\to \mathbb {B} )),\end{matrix}}}$

in which propositions ${\displaystyle F:\mathbb {B} ^{n}\to \mathbb {B} \!}$ are reconsidered as “proposition fields” ${\displaystyle {\boldsymbol {\iota }}F:\mathbb {B} ^{j}\to (\mathbb {B} ^{k}\to \mathbb {B} ),\!}$ where ${\displaystyle n=j+k.\!}$ This gives rise to “n choose j” renditions of the isomorphism ${\displaystyle {\boldsymbol {\iota }}\!}$ for each rendering of the original alphabet into a ${\displaystyle j\!}$-set and a ${\displaystyle k\!}$-set.

In the conception of an individual analytic form the set of variables is divided into two parts and the sense of each proposition is apportioned accordingly. The initial set of variables is used to sweep out a range of partial interpretations and thus to set the context of analysis. The remaining set is left uninterpreted and kept embodied in propositional coefficients which are associated with the “loci” (the points or topics) of this conceptual framework.

In appearance, the expression of the analytic form is made up of coefficient propositions on a subset of the variables taking their places next to partial interpretations over the complementary set. This may seem like a purely superficial change, since the analytic expansion preserves logical equivalence with the original proposition, but the good of a particular form for a particular situation lies precisely in the facility of its appearance being unaffected by its logical substance. Its choice and use are matters of pragmatic relevance and suitability to a purpose. This means that the worth of a particular form for a problematic situation can only be judged after the fact of its being applied, by the contribution it makes toward clarifying an opportune constellation of propositions, those which comprehend the situation and define the problem.

If we need to formalize the process, each isomorphism ${\displaystyle {\boldsymbol {\iota }}\!}$ is implicitly indexed by a subset ${\displaystyle U\!}$ of the original alphabet ${\displaystyle X,\!}$ but most often we consider only a single expansion at a time and can safely let context determine the sense.

Let ${\displaystyle X=\{x_{1},\ldots ,x_{m}\}\!}$ be our principal set of logical variables, and let ${\displaystyle X\!}$ be divided arbitrarily into a pair of subsets, which we may assume without loss of generality to form an initial segment ${\displaystyle U=\{x_{1},\ldots ,x_{j}\}\!}$ and a final segment ${\displaystyle V=\{x_{j+1},\ldots ,x_{m}\},\!}$ of cardinalities ${\displaystyle j\!}$ and ${\displaystyle k,\!}$ respectively, where ${\displaystyle m=j+k.\!}$ Corresponding to the choice of ${\displaystyle U\!}$ (which determines the complementary set ${\displaystyle V\!}$) there are a pair of isomorphisms,

 ${\displaystyle {\begin{matrix}{\boldsymbol {\iota }}_{U}:(X\to \mathbb {B} )&=&(U\times V\to \mathbb {B} )&\to &(U\to (V\to \mathbb {B} )),\\{\boldsymbol {\iota }}_{V}:(X\to \mathbb {B} )&=&(U\times V\to \mathbb {B} )&\to &(V\to (U\to \mathbb {B} )),\end{matrix}}}$

whose employments we describe by saying that the proposition ${\displaystyle F\!}$ can be factored into the proposition fields ${\displaystyle {\boldsymbol {\iota }}_{U}F\!}$ and ${\displaystyle {\boldsymbol {\iota }}_{V}F,\!}$ respectively. These factorizations are commonly expressed by means of the associated analytic forms, which determine a pair of logical expansions for each proposition ${\displaystyle F\!}$ in the universe ${\displaystyle X^{\bullet },\!}$

 ${\displaystyle {\begin{matrix}F(x)&=&F(u,v)&=&\sum _{u}{\boldsymbol {\varphi }}_{u}F(v)\cdot u,\\F(x)&=&F(u,v)&=&\sum _{v}{\boldsymbol {\varphi }}_{v}F(u)\cdot v.\end{matrix}}}$

### Elementary Examples

 The field contains the great elements, individuality, understanding, unmanifest nature, the eleven senses, and the five sense realms. — Bhagavad Gita, 13.5

A few examples of analytic expansions are appropriate here. The following selection exhibits the concepts and notation we need from previous discussions, indicates how we intend to adapt these materials to the present purpose, and provides a set of building blocks for future constructions.

To begin with a minimal example, the conjunction ${\displaystyle J(x,y)=x\cdot y\!}$ yields the partial expansion with respect to ${\displaystyle x,\!}$

 ${\displaystyle {\begin{array}{lrrrr}J(x,y)&=&J(1,y)\cdot x&+&J(0,y)\cdot (x),\\&=&{\boldsymbol {\varphi }}_{x}J(y)\cdot x&+&{\boldsymbol {\varphi }}_{(x)}J(y)\cdot (x),\\&=&y\cdot x&+&0\cdot (x),\end{array}}}$

the partial expansion with respect to ${\displaystyle y,\!}$

 ${\displaystyle {\begin{array}{lrrrr}J(x,y)&=&J(x,1)\cdot y&+&J(x,0)\cdot (y),\\&=&{\boldsymbol {\varphi }}_{y}J(x)\cdot y&+&{\boldsymbol {\varphi }}_{(y)}J(x)\cdot (y),\\&=&x\cdot y&+&0\cdot (y),\end{array}}}$

and the complete expansion with respect to ${\displaystyle x\!}$ and ${\displaystyle y,\!}$

 ${\displaystyle {\begin{array}{l*{8}{r}}J(x,y)&=&J(1,1)\cdot xy&+&J(1,0)\cdot x(y)&+&J(0,1)\cdot (x)y&+&J(0,0)\cdot (x)(y),\\&=&{\boldsymbol {\varphi }}_{xy}J\cdot xy&+&{\boldsymbol {\varphi }}_{x(y)}J\cdot x(y)&+&{\boldsymbol {\varphi }}_{(x)y}J\cdot (x)y&+&{\boldsymbol {\varphi }}_{(x)(y)}J\cdot (x)(y),\\&=&1\cdot xy&+&0\cdot x(y)&+&0\cdot (x)y&+&0\cdot (x)(y).\end{array}}}$

Notice how the results of the coefficient extraction (partial interpretation or partial evaluation) operators ${\displaystyle {\boldsymbol {\varphi }}_{\alpha },\!}$ where the index ${\displaystyle \alpha \!}$ refers to a singular proposition of the relevant category of discourse, could almost be defined in terms of the analytic form, namely, as the propositions occupying the designated places of the expansion. This becomes a real possibility if we have an independent way of developing the analytic form in a constructive or computational setting.

Finally, the reason we are taking such great pains to define all entities in terms of operators on names of expressions is because we wish to anticipate those circumstances of computation when we fail to have the full expressions available all the time, for instance, to evaluate by performing substitutions on.

Figure 3 uses the “bundle of boxes” style of venn diagram to illustrate the partial expansions of the conjunction ${\displaystyle J(x,y).\!}$

 ${\displaystyle {\text{Figure 3. Factorizations of Conjunction}}\!}$

As a slightly more complex example, consider the boolean function determined by the equality or biconditional ${\displaystyle I(x,y)=((x,y)).\!}$ This affords the partial expansion with respect to ${\displaystyle x,\!}$

 ${\displaystyle {\begin{array}{lrrrr}I(x,y)&=&y\cdot x&+&(y)\cdot (x),\end{array}}}$

the partial expansion with respect to ${\displaystyle y,\!}$

 ${\displaystyle {\begin{array}{lrrrr}I(x,y)&=&x\cdot y&+&(x)\cdot (y),\end{array}}}$

and the complete expansion with respect to ${\displaystyle x\!}$ and ${\displaystyle y,\!}$

 ${\displaystyle {\begin{array}{l*{8}{r}}I(x,y)&=&1\cdot xy&+&0\cdot x(y)&+&0\cdot (x)y&+&1\cdot (x)(y).\end{array}}}$

Figure 4 illustrates the partial expansions of the equality ${\displaystyle I(x,y).\!}$

 ${\displaystyle {\text{Figure 4. Factorizations of Equality}}\!}$

For our last example, the implication or conditional ${\displaystyle K(x,y)=(x(y))~\!}$ results in the partial expansion with respect to ${\displaystyle x,\!}$

 ${\displaystyle {\begin{array}{lrrrr}K(x,y)&=&y\cdot x&+&1\cdot (x),\end{array}}}$

the partial expansion with respect to ${\displaystyle y,\!}$

 ${\displaystyle {\begin{array}{lrrrr}K(x,y)&=&1\cdot y&+&(x)\cdot (y),\end{array}}}$

and the complete expansion with respect to ${\displaystyle x\!}$ and ${\displaystyle y,\!}$

 ${\displaystyle {\begin{array}{l*{8}{r}}K(x,y)&=&1\cdot xy&+&0\cdot x(y)&+&1\cdot (x)y&+&1\cdot (x)(y).\end{array}}}$

Figure 5 illustrates the partial expansions of the implication ${\displaystyle K(x,y).\!}$

 ${\displaystyle {\text{Figure 5. Factorizations of Implication}}\!}$

### Differential Enlargements

 Longing, hatred, happiness, suffering, bodily form, consciousness, resolve, thus is this field with its changes defined in summary. — Bhagavad Gita, 13.6

Another important operation treated in the Tools and Views paper was the differential enlargement of propositions. This a mapping of the type,

 ${\displaystyle \mathrm {E} :(U\to \mathbb {B} )\to (U\times \mathrm {d} U\to \mathbb {B} ),\!}$

whose action on any proposition ${\displaystyle f:U\to \mathbb {B} \!}$ is defined by an equation of the form,

 ${\displaystyle {\begin{matrix}\mathrm {E} f(x_{1},\ldots ,x_{n},\mathrm {d} x_{1},\ldots ,\mathrm {d} x_{n})&=&f(x_{1}+\mathrm {d} x_{1},\ldots ,x_{n}+\mathrm {d} x_{n}).\end{matrix}}}$

In the effort to comprehend what the differential enlargement of a proposition is telling us about the original proposition we usually parse it as a proposition field in either one of two obvious ways. If we factor the ordinary component out to the front we obtain the flow,

 ${\displaystyle {\begin{matrix}\Phi f&=&{\boldsymbol {\iota }}_{U}\mathrm {E} f:U\to (\mathrm {d} U\to \mathbb {B} ),\end{matrix}}}$

If we factor the differential component out to the front we obtain fray,

 ${\displaystyle {\begin{matrix}\Psi f&=&{\boldsymbol {\iota }}_{\mathrm {d} U}\mathrm {E} f:\mathrm {d} U\to (U\to \mathbb {B} ).\end{matrix}}}$

Evaluating the flow at points of the original universe, we discover the intentional attraction or influence of ${\displaystyle f\!}$ at each point,

 ${\displaystyle {\begin{matrix}T_{u}f&=&{\boldsymbol {\varphi }}_{u}\Phi f&=&{\boldsymbol {\varphi }}_{u}{\boldsymbol {\iota }}_{U}\mathrm {E} f:\mathrm {d} U\to \mathbb {B} .\end{matrix}}}$

Evaluating the fray along directions of the differential universe, we find the differential action or diffraction of ${\displaystyle f\!}$ in each direction,

 ${\displaystyle {\begin{matrix}T_{v}f&=&{\boldsymbol {\varphi }}_{v}\Psi f&=&{\boldsymbol {\varphi }}_{v}{\boldsymbol {\iota }}_{\mathrm {d} U}\mathrm {E} f:U\to \mathbb {B} .\end{matrix}}}$

Diffraction = differential factorization (or differential fractionation):

 ${\displaystyle {\begin{matrix}(\mathbb {B} ^{n}\times \mathbb {D} ^{n}\to \mathbb {B} )&\cong &(\mathbb {D} ^{n}\to (\mathbb {B} ^{n}\to \mathbb {B} ).\end{matrix}}}$

Between the twin perspectives afforded by these two ${\displaystyle {\boldsymbol {\iota }}\!}$'s we have what we need to synthesize a stereotactic grasp of the situation represented in a proposition.

## Differential Propositions and Transformation Groups

 Its hands and feet reach everywhere; its head and face see in every direction; hearing everything, it remains in the world, enveloping all. — Bhagavad Gita, 13.13

In this section we examine a number of relationships between differential operators and higher order propositions, together with the actions and characters of related transformation groups on the space of propositions.

### Differential Expansions

 Lacking all the sense organs, it shines in their qualities; unattached, it supports everything; without qualities, it enjoys them. — Bhagavad Gita, 13.14

On analogy with usage in ordinary calculus, we introduce the following terminology. Given two sets of logical features ${\displaystyle X\subseteq Y,\!}$ say

 ${\displaystyle X=\{x_{1},\ldots ,x_{n}\}\!}$ and ${\displaystyle Y=\{x_{1},\ldots ,x_{m}\},\!}$

a proposition ${\displaystyle F:Y\to \mathbb {B} \!}$ is called an infinitesimal with respect to the universe ${\displaystyle X,\!}$ written ${\displaystyle F\in \Upsilon (X),\!}$ if ${\displaystyle F\!}$ is false outside the region of the logical disjunction ${\displaystyle ((x_{1})(\ldots )(x_{n})),\!}$ in other words, if ${\displaystyle F=0\!}$ at the origin ${\displaystyle (x_{1})(\ldots )(x_{n})\!}$ of ${\displaystyle X.\!}$

 ${\displaystyle {\text{Figure 6. Enlargement of Conjunction}}\!}$

 ${\displaystyle {\text{Figure 7. Diffraction of Conjunction}}\!}$

 ${\displaystyle {\text{Figure 8. Enlargement of Equality}}\!}$

 ${\displaystyle {\text{Figure 9. Diffraction of Equality}}\!}$

 ${\displaystyle {\text{Figure 10. Enlargement of Implication}}\!}$

 ${\displaystyle {\text{Figure 11. Diffraction of Implication}}\!}$

### Partial Evaluations

 Outside and within all creatures, inanimate but still animate, too subtle to be known, it is far distant, yet near. — Bhagavad Gita, 13.15

 ${\displaystyle f\!}$ ${\displaystyle \mathrm {E} f\!}$ ${\displaystyle \mathrm {E} f\!}$ @ ${\displaystyle x\cdot y\!}$ ${\displaystyle \mathrm {E} f\!}$ @ ${\displaystyle x\cdot (y)\!}$ ${\displaystyle \mathrm {E} f\!}$ @ ${\displaystyle (x)\cdot y\!}$ ${\displaystyle \mathrm {E} f\!}$ @ ${\displaystyle (x)(y)\!}$ ${\displaystyle f_{0}\!}$ ${\displaystyle ()\!}$ ${\displaystyle ()\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle f_{1}\!}$ ${\displaystyle (x)(y)\!}$ ${\displaystyle ((x,\mathrm {d} x))((y,\mathrm {d} y))\!}$ ${\displaystyle \mathrm {d} x~\mathrm {d} y\!}$ ${\displaystyle \mathrm {d} x(\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x)\mathrm {d} y\!}$ ${\displaystyle (\mathrm {d} x)(\mathrm {d} y)\!}$ ${\displaystyle f_{2}\!}$ ${\displaystyle (x)y\!}$ ${\displaystyle ((x,\mathrm {d} x))(y,\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} x(\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} x~\mathrm {d} y\!}$ ${\displaystyle (\mathrm {d} x)(\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x)\mathrm {d} y\!}$ ${\displaystyle f_{4}\!}$ ${\displaystyle x(y)\!}$ ${\displaystyle (x,\mathrm {d} x)((y,\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x)\mathrm {d} y\!}$ ${\displaystyle (\mathrm {d} x)(\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} x~\mathrm {d} y\!}$ ${\displaystyle \mathrm {d} x(\mathrm {d} y)\!}$ ${\displaystyle f_{8}\!}$ ${\displaystyle xy\!}$ ${\displaystyle (x,\mathrm {d} x)(y,\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x)(\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x)\mathrm {d} y\!}$ ${\displaystyle \mathrm {d} x(\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} x~\mathrm {d} y\!}$ ${\displaystyle f_{3}\!}$ ${\displaystyle (x)\!}$ ${\displaystyle ((x,\mathrm {d} x))\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle (\mathrm {d} x)\!}$ ${\displaystyle (\mathrm {d} x)\!}$ ${\displaystyle f_{12}\!}$ ${\displaystyle x\!}$ ${\displaystyle (x,\mathrm {d} x)\!}$ ${\displaystyle (\mathrm {d} x)\!}$ ${\displaystyle (\mathrm {d} x)\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle f_{6}\!}$ ${\displaystyle (x,y)\!}$ ${\displaystyle ((x,\mathrm {d} x),(y,\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x,\mathrm {d} y)\!}$ ${\displaystyle ((\mathrm {d} x,\mathrm {d} y))\!}$ ${\displaystyle ((\mathrm {d} x,\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x,\mathrm {d} y)\!}$ ${\displaystyle f_{9}\!}$ ${\displaystyle ((x,y))\!}$ ${\displaystyle (((x,\mathrm {d} x),(y,\mathrm {d} y)))~\!}$ ${\displaystyle ((\mathrm {d} x,\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x,\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x,\mathrm {d} y)\!}$ ${\displaystyle ((\mathrm {d} x,\mathrm {d} y))\!}$ ${\displaystyle f_{5}\!}$ ${\displaystyle (y)\!}$ ${\displaystyle ((y,\mathrm {d} y))\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle (\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle (\mathrm {d} y)\!}$ ${\displaystyle f_{10}\!}$ ${\displaystyle y\!}$ ${\displaystyle (y,\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle (\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle f_{7}\!}$ ${\displaystyle (xy)\!}$ ${\displaystyle ((x,\mathrm {d} x)(y,\mathrm {d} y))\!}$ ${\displaystyle ((\mathrm {d} x)(\mathrm {d} y))\!}$ ${\displaystyle ((\mathrm {d} x)\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x(\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x~\mathrm {d} y)\!}$ ${\displaystyle f_{11}\!}$ ${\displaystyle (x(y))\!}$ ${\displaystyle ((x,\mathrm {d} x)((y,\mathrm {d} y)))\!}$ ${\displaystyle ((\mathrm {d} x)\mathrm {d} y)\!}$ ${\displaystyle ((\mathrm {d} x)(\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x~\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x(\mathrm {d} y))\!}$ ${\displaystyle f_{13}\!}$ ${\displaystyle ((x)y)\!}$ ${\displaystyle (((x,\mathrm {d} x))(y,\mathrm {d} y))~\!}$ ${\displaystyle (\mathrm {d} x(\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x~\mathrm {d} y)\!}$ ${\displaystyle ((\mathrm {d} x)(\mathrm {d} y))\!}$ ${\displaystyle ((\mathrm {d} x)\mathrm {d} y)\!}$ ${\displaystyle f_{14}\!}$ ${\displaystyle ((x)(y))\!}$ ${\displaystyle (((x,\mathrm {d} x))((y,\mathrm {d} y)))\!}$ ${\displaystyle (\mathrm {d} x~\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x(\mathrm {d} y))\!}$ ${\displaystyle ((\mathrm {d} x)\mathrm {d} y)\!}$ ${\displaystyle ((\mathrm {d} x)(\mathrm {d} y))\!}$ ${\displaystyle f_{15}\!}$ ${\displaystyle (())\!}$ ${\displaystyle (())\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$

### Group Actions

 Undivided, it seems divided among creatures; understood as their sustainer, it devours and creates them. — Bhagavad Gita, 13.16

 ${\displaystyle f\!}$ ${\displaystyle \mathrm {E} f\!}$ ${\displaystyle T_{11}f\!}$ ${\displaystyle T_{10}f\!}$ ${\displaystyle T_{01}f\!}$ ${\displaystyle T_{00}f\!}$ ${\displaystyle x=x_{1}\!}$ ${\displaystyle y=x_{2}\!}$ ${\displaystyle \mathrm {sub} _{i}[(x_{i}+\mathrm {d} x_{i})/x_{i}]F\!}$ ${\displaystyle \mathrm {E} f\!}$ @ ${\displaystyle \mathrm {d} x\cdot \mathrm {d} y\!}$ ${\displaystyle \mathrm {E} f\!}$ @ ${\displaystyle \mathrm {d} x\cdot (\mathrm {d} y)\!}$ ${\displaystyle \mathrm {E} f\!}$ @ ${\displaystyle (\mathrm {d} x)\cdot \mathrm {d} y\!}$ ${\displaystyle \mathrm {E} f\!}$ @ ${\displaystyle (\mathrm {d} x)(\mathrm {d} y)\!}$ ${\displaystyle f_{0}\!}$ ${\displaystyle ()\!}$ ${\displaystyle ()\!}$ ${\displaystyle ()\!}$ ${\displaystyle ()\!}$ ${\displaystyle ()\!}$ ${\displaystyle ()\!}$ ${\displaystyle f_{1}\!}$ ${\displaystyle (x)(y)\!}$ ${\displaystyle ((x,\mathrm {d} x))((y,\mathrm {d} y))\!}$ ${\displaystyle xy\!}$ ${\displaystyle x(y)\!}$ ${\displaystyle (x)y\!}$ ${\displaystyle (x)(y)\!}$ ${\displaystyle f_{2}\!}$ ${\displaystyle (x)y\!}$ ${\displaystyle ((x,\mathrm {d} x))(y,\mathrm {d} y)\!}$ ${\displaystyle x(y)\!}$ ${\displaystyle xy\!}$ ${\displaystyle (x)(y)\!}$ ${\displaystyle (x)y\!}$ ${\displaystyle f_{4}\!}$ ${\displaystyle x(y)\!}$ ${\displaystyle (x,\mathrm {d} x)((y,\mathrm {d} y))\!}$ ${\displaystyle (x)y\!}$ ${\displaystyle (x)(y)\!}$ ${\displaystyle xy\!}$ ${\displaystyle x(y)\!}$ ${\displaystyle f_{8}\!}$ ${\displaystyle xy\!}$ ${\displaystyle (x,\mathrm {d} x)(y,\mathrm {d} y)\!}$ ${\displaystyle (x)(y)\!}$ ${\displaystyle (x)y\!}$ ${\displaystyle x(y)\!}$ ${\displaystyle xy\!}$ ${\displaystyle f_{3}\!}$ ${\displaystyle (x)\!}$ ${\displaystyle ((x,\mathrm {d} x))\!}$ ${\displaystyle x\!}$ ${\displaystyle x\!}$ ${\displaystyle (x)\!}$ ${\displaystyle (x)\!}$ ${\displaystyle f_{12}\!}$ ${\displaystyle x\!}$ ${\displaystyle (x,\mathrm {d} x)\!}$ ${\displaystyle (x)\!}$ ${\displaystyle (x)\!}$ ${\displaystyle x\!}$ ${\displaystyle x\!}$ ${\displaystyle f_{6}\!}$ ${\displaystyle (x,y)\!}$ ${\displaystyle ((x,\mathrm {d} x),(y,\mathrm {d} y))\!}$ ${\displaystyle (x,y)\!}$ ${\displaystyle ((x,y))\!}$ ${\displaystyle ((x,y))\!}$ ${\displaystyle (x,y)\!}$ ${\displaystyle f_{9}\!}$ ${\displaystyle ((x,y))\!}$ ${\displaystyle (((x,\mathrm {d} x),(y,\mathrm {d} y)))~\!}$ ${\displaystyle ((x,y))\!}$ ${\displaystyle (x,y)\!}$ ${\displaystyle (x,y)\!}$ ${\displaystyle ((x,y))\!}$ ${\displaystyle f_{5}\!}$ ${\displaystyle (y)\!}$ ${\displaystyle ((y,\mathrm {d} y))\!}$ ${\displaystyle y\!}$ ${\displaystyle (y)\!}$ ${\displaystyle y\!}$ ${\displaystyle (y)\!}$ ${\displaystyle f_{10}\!}$ ${\displaystyle y\!}$ ${\displaystyle (y,\mathrm {d} y)\!}$ ${\displaystyle (y)\!}$ ${\displaystyle y\!}$ ${\displaystyle (y)\!}$ ${\displaystyle y\!}$ ${\displaystyle f_{7}\!}$ ${\displaystyle (xy)\!}$ ${\displaystyle ((x,\mathrm {d} x)(y,\mathrm {d} y))\!}$ ${\displaystyle ((x)(y))\!}$ ${\displaystyle ((x)y)\!}$ ${\displaystyle (x(y))\!}$ ${\displaystyle (xy)\!}$ ${\displaystyle f_{11}\!}$ ${\displaystyle (x(y))\!}$ ${\displaystyle ((x,\mathrm {d} x)((y,\mathrm {d} y)))\!}$ ${\displaystyle ((x)y)\!}$ ${\displaystyle ((x)(y))\!}$ ${\displaystyle (xy)\!}$ ${\displaystyle (x(y))\!}$ ${\displaystyle f_{13}\!}$ ${\displaystyle ((x)y)\!}$ ${\displaystyle (((x,\mathrm {d} x))(y,\mathrm {d} y))~\!}$ ${\displaystyle (x(y))\!}$ ${\displaystyle (xy)\!}$ ${\displaystyle ((x)(y))\!}$ ${\displaystyle ((x)y)\!}$ ${\displaystyle f_{14}\!}$ ${\displaystyle ((x)(y))\!}$ ${\displaystyle (((x,\mathrm {d} x))((y,\mathrm {d} y)))\!}$ ${\displaystyle (xy)\!}$ ${\displaystyle (x(y))\!}$ ${\displaystyle ((x)y)\!}$ ${\displaystyle ((x)(y))\!}$ ${\displaystyle f_{15}\!}$ ${\displaystyle (())\!}$ ${\displaystyle (())\!}$ ${\displaystyle (())\!}$ ${\displaystyle (())\!}$ ${\displaystyle (())\!}$ ${\displaystyle (())\!}$ ${\displaystyle {\text{Total Number of Fixed Points:}}\!}$ ${\displaystyle 4\!}$ ${\displaystyle 4\!}$ ${\displaystyle 4\!}$ ${\displaystyle 16\!}$

### Derivations

 The light of lights beyond darkness it is called; knowledge attained by knowledge, fixed in the heart of everyone. — Bhagavad Gita, 13.17

 ${\displaystyle \mathrm {D} f=f+\mathrm {E} f\!}$ ${\displaystyle \mathrm {D} f~\!}$ @ ${\displaystyle x\cdot y\!}$ ${\displaystyle \mathrm {D} f~\!}$ @ ${\displaystyle x\cdot (y)\!}$ ${\displaystyle \mathrm {D} f~\!}$ @ ${\displaystyle (x)\cdot y\!}$ ${\displaystyle \mathrm {D} f~\!}$ @ ${\displaystyle (x)(y)\!}$ ${\displaystyle f_{0}\!}$ ${\displaystyle ()+()\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle f_{1}\!}$ ${\displaystyle (x)(y)+((x,\mathrm {d} x))((y,\mathrm {d} y))\!}$ ${\displaystyle \mathrm {d} x~\mathrm {d} y\!}$ ${\displaystyle \mathrm {d} x(\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x)\mathrm {d} y\!}$ ${\displaystyle ((\mathrm {d} x)(\mathrm {d} y))\!}$ ${\displaystyle f_{2}\!}$ ${\displaystyle (x)y+((x,\mathrm {d} x))(y,\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} x(\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} x~\mathrm {d} y\!}$ ${\displaystyle ((\mathrm {d} x)(\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x)\mathrm {d} y\!}$ ${\displaystyle f_{4}\!}$ ${\displaystyle x(y)+(x,\mathrm {d} x)((y,\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x)\mathrm {d} y\!}$ ${\displaystyle ((\mathrm {d} x)(\mathrm {d} y))\!}$ ${\displaystyle \mathrm {d} x~\mathrm {d} y\!}$ ${\displaystyle \mathrm {d} x(\mathrm {d} y)\!}$ ${\displaystyle f_{8}\!}$ ${\displaystyle xy+(x,\mathrm {d} x)(y,\mathrm {d} y)\!}$ ${\displaystyle ((\mathrm {d} x)(\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x)\mathrm {d} y\!}$ ${\displaystyle \mathrm {d} x(\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} x~\mathrm {d} y\!}$ ${\displaystyle f_{3}\!}$ ${\displaystyle (x)+((x,\mathrm {d} x))\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle f_{12}\!}$ ${\displaystyle x+(x,\mathrm {d} x)\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle f_{6}\!}$ ${\displaystyle (x,y)+((x,\mathrm {d} x),(y,\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x,\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x,\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x,\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x,\mathrm {d} y)\!}$ ${\displaystyle f_{9}\!}$ ${\displaystyle ((x,y))+(((x,\mathrm {d} x),(y,\mathrm {d} y)))\!}$ ${\displaystyle (\mathrm {d} x,\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x,\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x,\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x,\mathrm {d} y)\!}$ ${\displaystyle f_{5}\!}$ ${\displaystyle (y)+((y,\mathrm {d} y))\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle f_{10}\!}$ ${\displaystyle y+(y,\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle f_{7}\!}$ ${\displaystyle (xy)+((x,\mathrm {d} x)(y,\mathrm {d} y))\!}$ ${\displaystyle ((\mathrm {d} x)(\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x)\mathrm {d} y\!}$ ${\displaystyle \mathrm {d} x(\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} x~\mathrm {d} y\!}$ ${\displaystyle f_{11}\!}$ ${\displaystyle (x(y))+((x,\mathrm {d} x)((y,\mathrm {d} y)))\!}$ ${\displaystyle (\mathrm {d} x)\mathrm {d} y\!}$ ${\displaystyle ((\mathrm {d} x)(\mathrm {d} y))\!}$ ${\displaystyle \mathrm {d} x~\mathrm {d} y\!}$ ${\displaystyle \mathrm {d} x(\mathrm {d} y)\!}$ ${\displaystyle f_{13}\!}$ ${\displaystyle ((x)y)+(((x,\mathrm {d} x))(y,\mathrm {d} y))\!}$ ${\displaystyle \mathrm {d} x(\mathrm {d} y)\!}$ ${\displaystyle \mathrm {d} x~\mathrm {d} y\!}$ ${\displaystyle ((\mathrm {d} x)(\mathrm {d} y))\!}$ ${\displaystyle (\mathrm {d} x)\mathrm {d} y\!}$ ${\displaystyle f_{14}\!}$ ${\displaystyle ((x)(y))+(((x,\mathrm {d} x))((y,\mathrm {d} y)))\!}$ ${\displaystyle \mathrm {d} x~\mathrm {d} y\!}$ ${\displaystyle \mathrm {d} x(\mathrm {d} y)\!}$ ${\displaystyle (\mathrm {d} x)\mathrm {d} y\!}$ ${\displaystyle ((\mathrm {d} x)(\mathrm {d} y))\!}$ ${\displaystyle f_{15}\!}$ ${\displaystyle (())+(())\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$

## Generalized Projections : Analytic Operators and Fields of Inquiry

 Arjuna, know that anything inanimate or alive with motion is born from the union of the field and its knower. — Bhagavad Gita, 13.26

Projections, Dissections, Distortions, Perspectives (Outlooks, Opinions, Local Views),

Partial Derivatives, Fields of Inquiry, Surveys, Test Fields, Textures, Cultures, Biases, Warps.

Relation to questions of value-free inquiry and inquiry into values, to what extent these projects are possible or approachable from the stance of a concrete interpreter.

Analytic operators or derivations, all of type ${\displaystyle q:(\mathbb {B} ^{n}\to \mathbb {B} )\to (\mathbb {B} ^{n}\to \mathbb {B} ).\!}$ Includes projections and partial derivatives. Equivalence to proposition fields, question fields, fields of inquiry, as follows:

 ${\displaystyle Q:\mathbb {B} ^{n}\to (F:(\mathbb {B} ^{n}\to \mathbb {B} )\to \mathbb {B} ).\!}$

Note relation between operator symmetry and dimension reduction. When the question field ${\displaystyle Q\!}$ is based on a symmetric operation, ${\displaystyle f(x,y)=f(y,x),\!}$ then the corresponding analytic operator ${\displaystyle q\!}$ may be viewed under a dimension-reducing type as ${\displaystyle q:(\mathbb {B} ^{n}\to \mathbb {B} )\to (\mathbb {B} ^{n-1}\to \mathbb {B} ).\!}$

 ${\displaystyle f\!}$ ${\displaystyle \theta f_{xy1}\!}$ ${\displaystyle \theta f_{xy0}\!}$ ${\displaystyle \theta f_{1yz}\!}$ ${\displaystyle \theta f_{0yz}\!}$ ${\displaystyle \theta f_{x1z}\!}$ ${\displaystyle \theta f_{x0z}\!}$ ${\displaystyle f_{0}\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle (z)\!}$ ${\displaystyle (z)\!}$ ${\displaystyle (z)\!}$ ${\displaystyle (z)\!}$ ${\displaystyle f_{1}\!}$ ${\displaystyle (x)(y)\!}$ ${\displaystyle ((x)(y))\!}$ ${\displaystyle (z)\!}$ ${\displaystyle (y,z)\!}$ ${\displaystyle (z)\!}$ ${\displaystyle (x,z)\!}$ ${\displaystyle f_{2}\!}$ ${\displaystyle (x)y\!}$ ${\displaystyle ((x)y)\!}$ ${\displaystyle (z)\!}$ ${\displaystyle ((y,z))\!}$ ${\displaystyle (x,z)\!}$ ${\displaystyle (z)\!}$ ${\displaystyle f_{4}\!}$ ${\displaystyle x(y)\!}$ ${\displaystyle (x(y))\!}$ ${\displaystyle (y,z)\!}$ ${\displaystyle (z)\!}$ ${\displaystyle (z)\!}$ ${\displaystyle ((x,z))\!}$ ${\displaystyle f_{8}\!}$ ${\displaystyle xy\!}$ ${\displaystyle (xy)\!}$ ${\displaystyle ((y,z))\!}$ ${\displaystyle (z)\!}$ ${\displaystyle ((x,z))\!}$ ${\displaystyle (z)\!}$ ${\displaystyle f_{3}\!}$ ${\displaystyle (x)\!}$ ${\displaystyle x\!}$ ${\displaystyle (z)\!}$ ${\displaystyle z\!}$ ${\displaystyle (x,z)\!}$ ${\displaystyle (x,z)\!}$ ${\displaystyle f_{12}\!}$ ${\displaystyle x\!}$ ${\displaystyle (x)\!}$ ${\displaystyle z\!}$ ${\displaystyle (z)\!}$ ${\displaystyle ((x,z))\!}$ ${\displaystyle ((x,z))\!}$ ${\displaystyle f_{6}\!}$ ${\displaystyle (x,y)\!}$ ${\displaystyle ((x,y))\!}$ ${\displaystyle (y,z)\!}$ ${\displaystyle ((y,z))\!}$ ${\displaystyle (x,z)\!}$ ${\displaystyle ((x,z))\!}$ ${\displaystyle f_{9}\!}$ ${\displaystyle ((x,y))\!}$ ${\displaystyle (x,y)\!}$ ${\displaystyle ((y,z))\!}$ ${\displaystyle (y,z)\!}$ ${\displaystyle ((x,z))\!}$ ${\displaystyle (x,z)\!}$ ${\displaystyle f_{5}\!}$ ${\displaystyle (y)\!}$ ${\displaystyle y\!}$ ${\displaystyle (y,z)\!}$ ${\displaystyle (y,z)\!}$ ${\displaystyle (z)\!}$ ${\displaystyle z\!}$ ${\displaystyle f_{10}\!}$ ${\displaystyle y\!}$ ${\displaystyle (y)\!}$ ${\displaystyle ((y,z))\!}$ ${\displaystyle ((y,z))\!}$ ${\displaystyle z\!}$ ${\displaystyle (z)\!}$ ${\displaystyle f_{7}\!}$ ${\displaystyle (xy)\!}$ ${\displaystyle xy\!}$ ${\displaystyle (y,z)\!}$ ${\displaystyle z\!}$ ${\displaystyle (x,z)\!}$ ${\displaystyle z\!}$ ${\displaystyle f_{11}\!}$ ${\displaystyle (x(y))\!}$ ${\displaystyle x(y)\!}$ ${\displaystyle ((y,z))\!}$ ${\displaystyle z\!}$ ${\displaystyle z\!}$ ${\displaystyle (x,z)\!}$ $\displaystyle f_{13}\!$ ${\displaystyle ((x)y)\!}$ ${\displaystyle (x)y\!}$ ${\displaystyle z\!}$ ${\displaystyle (y,z)\!}$ ${\displaystyle ((x,z))\!}$ ${\displaystyle z\!}$ ${\displaystyle f_{14}\!}$ ${\displaystyle ((x)(y))\!}$ ${\displaystyle (x)(y)\!}$ ${\displaystyle z\!}$ ${\displaystyle ((y,z))\!}$ $\displaystyle z\!$ ${\displaystyle ((x,z))\!}$ ${\displaystyle f_{15}\!}$ $\displaystyle 1\!$ $\displaystyle 0\!$ $\displaystyle z\!$ $\displaystyle z\!$ $\displaystyle z\!$ $\displaystyle z\!$

## References

[Bhag] The Bhagavad-Gita : Krishna's Counsel in Time of War, B.S. Miller (trans.), Bantam Books, New York, NY, 1986.

## Fragments

Note. This part needs work. It looks like I was trying to get $\displaystyle \frac{\mathrm{d}x}{\mathrm{d}y}\!$ and $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}\!$ relative to the region where ${\displaystyle f\!}$ is true. But the fragmentary notes I have found so far are bedeviled by a confused mix of conceptual, notational, and typographical errors. In ordinary real-value calculus implicit differentiation of $\displaystyle f(x, y) = 1\!$ gives the following formulas:

 ${\displaystyle {\begin{matrix}{\mathrm {d} _{x}y}={\frac {\mathrm {d} y}{\mathrm {d} x}}=-{\frac {f_{x}}{f_{y}}}=-{\frac {\partial _{x}f}{\partial _{y}f}}=-{\frac {{\partial f}/{\partial x}}{{\partial f}/{\partial y}}}&\qquad &{\mathrm {d} _{y}x}={\frac {\mathrm {d} x}{\mathrm {d} y}}=-{\frac {f_{y}}{f_{x}}}=-{\frac {\partial _{y}f}{\partial _{x}f}}=-{\frac {{\partial f}/{\partial y}}{{\partial f}/{\partial x}}}\end{matrix}}\!}$

The minus signs can be dropped in the boolean context but taking quotients is still a problem. If anything analogous to the preceding formulas does go through, it looks like I can correct the notation in the last two column heads to give the following table:

 ${\displaystyle f\!}$ ${\displaystyle {\frac {\partial f}{\partial x}}\!}$ ${\displaystyle {\frac {\partial f}{\partial y}}\!}$ ${\displaystyle {\begin{matrix}\mathrm {d} f=\\[2pt]\partial _{x}f\cdot \mathrm {d} x~+~\partial _{y}f\cdot \mathrm {d} y\end{matrix}}\!}$ ${\displaystyle {\begin{matrix}{\mathrm {d} _{x}y}|f=\\[2pt]{\partial _{x}f}/{\partial _{y}f}\end{matrix}}\!}$ ${\displaystyle {\begin{matrix}{\mathrm {d} _{y}x}|f=\\[2pt]{\partial _{y}f}/{\partial _{x}f}\end{matrix}}\!}$ ${\displaystyle f_{0}\!}$ ${\displaystyle ()\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0/0\!}$ ${\displaystyle 0/0\!}$ ${\displaystyle f_{1}\!}$ ${\displaystyle (x)(y)\!}$ ${\displaystyle (y)\!}$ ${\displaystyle (x)\!}$ ${\displaystyle (y)~\mathrm {d} x~+~(x)~\mathrm {d} y\!}$ ${\displaystyle (y)/(x)\!}$ ${\displaystyle (x)/(y)\!}$ ${\displaystyle f_{2}\!}$ ${\displaystyle (x)y\!}$ ${\displaystyle y\!}$ ${\displaystyle (x)\!}$ ${\displaystyle y~\mathrm {d} x~+~(x)~\mathrm {d} y\!}$ ${\displaystyle y/(x)\!}$ ${\displaystyle (x)/y\!}$ ${\displaystyle f_{4}\!}$ ${\displaystyle x(y)\!}$ ${\displaystyle (y)\!}$ ${\displaystyle x\!}$ ${\displaystyle (y)~\mathrm {d} x~+~x~\mathrm {d} y\!}$ ${\displaystyle (y)/x\!}$ ${\displaystyle x/(y)\!}$ ${\displaystyle f_{8}\!}$ ${\displaystyle xy\!}$ ${\displaystyle y\!}$ ${\displaystyle x\!}$ $\displaystyle y ~ \mathrm{d}x ~+~ x ~ \mathrm{d}y\!$ $\displaystyle y/x\!$ ${\displaystyle x/y\!}$ ${\displaystyle f_{3}\!}$ $\displaystyle (x)\!$ ${\displaystyle 1\!}$ ${\displaystyle 0\!}$ $\displaystyle \mathrm{d}x\!$ $\displaystyle 1/0\!$ $\displaystyle 0\!$ ${\displaystyle f_{12}\!}$ ${\displaystyle x\!}$ ${\displaystyle 1\!}$ ${\displaystyle 0\!}$ ${\displaystyle \mathrm {d} x\!}$ ${\displaystyle 1/0\!}$ ${\displaystyle 0\!}$ ${\displaystyle f_{6}\!}$ ${\displaystyle (x,y)\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle \mathrm {d} x+\mathrm {d} y\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle f_{9}\!}$ ${\displaystyle ((x,y))\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle \mathrm {d} x+\mathrm {d} y\!}$ ${\displaystyle 1\!}$ ${\displaystyle 1\!}$ ${\displaystyle f_{5}\!}$ ${\displaystyle (y)\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1\!}$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1/0\!}$ ${\displaystyle f_{10}\!}$ ${\displaystyle y\!}$ ${\displaystyle 0\!}$ $\displaystyle 1\!$ ${\displaystyle \mathrm {d} y\!}$ ${\displaystyle 0\!}$ ${\displaystyle 1/0\!}$ ${\displaystyle f_{7}\!}$ ${\displaystyle (xy)\!}$ ${\displaystyle y\!}$ ${\displaystyle x\!}$ ${\displaystyle y~\mathrm {d} x~+~x~\mathrm {d} y\!}$ ${\displaystyle y/x\!}$ ${\displaystyle x/y\!}$ ${\displaystyle f_{11}\!}$ ${\displaystyle (x(y))\!}$ ${\displaystyle (y)\!}$ ${\displaystyle x\!}$ ${\displaystyle (y)~\mathrm {d} x~+~x~\mathrm {d} y\!}$ $\displaystyle (y)/x\!$ ${\displaystyle x/(y)\!}$ $\displaystyle f_{13}\!$ ${\displaystyle ((x)y)\!}$ ${\displaystyle y\!}$ ${\displaystyle (x)\!}$ ${\displaystyle y~\mathrm {d} x~+~(x)~\mathrm {d} y\!}$ ${\displaystyle y/(x)\!}$ ${\displaystyle (x)/y\!}$ ${\displaystyle f_{14}\!}$ ${\displaystyle ((x)(y))\!}$ ${\displaystyle (y)\!}$ ${\displaystyle (x)\!}$ ${\displaystyle (y)~\mathrm {d} x~+~(x)~\mathrm {d} y\!}$ ${\displaystyle (y)/(x)\!}$ ${\displaystyle (x)/(y)\!}$ ${\displaystyle f_{15}\!}$ ${\displaystyle (())\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0\!}$ ${\displaystyle 0/0\!}$ ${\displaystyle 0/0\!}$

 ${\displaystyle {\text{Figure B. Enlargements of Conjunction}}\!}$

 ${\displaystyle {\text{Figure C. Differentials of Conjunction}}\!}$

## Work Area

Inquiry Driven System as Uncertainty Control System

Types of Inquiry Arising from Types of Uncertainty

1. Uncertainty about what is true. Logical.

2. Uncertainty about what to do. Ethical.

3. Uncertainty about what to hope. Esthetic.

Do the Following Fall Under the Preceding?

Uncertainty about what interpretant to form.
Uncertainty about how to represent the present.
Uncertainty about how to approach the moment.
Uncertainty about what rheme to impose on the ration.
Uncertainty about what role to take up and cast others in the play.
Uncertainty about how to express, convey, carry, bear the moment.
Uncertainty about what relation to form from elements of the situation.
(Interpretant = equivalent, approximate, compensatory, substitute, or proxy sign.)

## Sources

Sanjaya, tell me what my sons
and the sons of Pandu did when they met,
wanting to battle on the field of Kuru,
on the field of sacred duty?
— Dhritarashtra, Bhagavad Gita, 1.1

Guarded by Bhishma, the strength
of our army is without limit;
but the strength of their army,
guarded by Bhima, is limited.
— Duryodhana, Bhagavad Gita, 1.10

Conches and kettledrums,
cymbals, tabors, and trumpets
were sounded at once
and the din of tumult arose.
— Sanjaya, Bhagavad Gita, 1.13

Arjuna, his war flag a rampant monkey,
saw Dhritarashtra's sons assembled
as weapons were ready to clash,
and he lifted his bow.
— Sanjaya, Bhagavad Gita, 1.20

Nothing of nonbeing comes to be,
nor does being cease to exist;
the boundary between these two
is seen by men who see reality.
— Krishna, Bhagavad Gita, 2.16

Indestructible is the presence
that pervades all this;
no one can destroy
this unchanging reality.

Creatures are unmanifest in origin,
manifest in the midst of life,
and unmanifest again in the end.
Since this is so, why do you lament?

Arjuna, the realm of sacred lore
is nature — beyond its triad of qualities,
dualities, and mundane rewards,
be forever lucid, alive to your self.

For the discerning priest,
all of sacred lore
has no more value than a well
when water flows everywhere.

Be intent on action,
not on the fruits of action;
avoid attraction to the fruits
and attachment to inaction!

Perform actions, firm in discipline,
relinquishing attachment;
be impartial to failure and success —
this equanimity is called discipline.

When it is night for all creatures,
a master of restraint is awake;
when they are awake, it is night
for the sage who sees reality.

No wise man disturbs the understanding
of ignorant men attached to action;
he should inspire them,
performing all actions with discipline.

Actions are all effected
by the qualities of nature;
but deluded by individuality,
the self thinks, “I am the actor.”

When he can discriminate
the actions of nature's qualities
and think, “The qualities depend
on other qualities,” he is detached.

Attraction and hatred are poised
in the object of every sense experience;
a man must not fall prey
to these two brigands lurking on his path!

The senses, mind, and understanding
are said to harbor desire;
with these desire obscures knowledge
and confounds the embodied self.

What is action? What is inaction?
Even the poets were confused —
what I shall teach you of action
will free you from misfortune.

When ignorance is destroyed
by knowledge of the self,
then, like the sun, knowledge
illumines ultimate reality.

That becomes their understanding,
their self, their basis, and their goal,
and they reach a state beyond return,
their sin dispelled by knowledge.

Arming himself with discipline,
seeing everything with an equal eye,
he sees the self in all creatures
and all creatures in the self.

When he sees identity in everything,
whether joy or suffering,
through analogy with the self,
he is deemed a man of pure discipline.

Doomed by his double failure,
is he not like a cloud split apart,
unsettled, deluded on the path
of the infinite spirit?
— Arjuna, Bhagavad Gita, 6.38

Arjuna, he does not suffer
doom in this world or the next;
any man who acts with honor
cannot go the wrong way, my friend.
— Krishna, Bhagavad Gita, 6.40

Nothing is higher than I am;
Arjuna, all that exists
is woven on me,
like a web of pearls on thread.

I am the taste in water, Arjuna,
the light in the moon and sun,
OM resonant in all sacred lore,
the sound in space, valor in men.

Beyond this unmanifest nature
is another unmanifest existence,
a timeless being that does not perish
when all creatures perish.

It is called eternal unmanifest nature,
what men call the highest way,
the goal from which they do not return;
this highest realm is mine.

It is man's highest spirit,
won by singular devotion, Arjuna,
in whom creatures rest
and the whole universe extends.

These bright and dark pathways
are deemed constant for the universe;
by one, a man escapes rebirth;
by the other, he is born again.

The whole universe is pervaded
by my unmanifest form;
all creatures exist in me,
but I do not exist in them.

Behold the power of my discipline;
these creatures are really not in me;
my self quickens creatures,
sustaining them without being in them.

Just as the wide-moving wind
is constantly present in space,
so all creatures exist in me;
understand it to be so!

I am heat that withholds
and sends down the rains;
I am immortality and death;
both being and nonbeing am I.

I am the self abiding
in the heart of all creatures;
I am their beginning,
their middle, and their end.

I am the endless cosmic serpent,
the lord of all sea creatures;
I am chief of the ancestral fathers;
of restraints, I am death.

I am the pious son of demons;
of measures, I am time;
I am the lion among wild animals,
the eagle among birds.

I am the vowel a of the syllabary,
the pairing of words in a compound;
I am indestructible time,
the creator facing everywhere at once.

I am death the destroyer of all,
the source of what will be,
the feminine powers: fame, fortune, speech,
memory, intelligence, resolve, patience.

Whatever is powerful, lucid,
splendid, or invulnerable
has its source in a fragment
of my brilliance.

What use is so much knowledge
to you, Arjuna?
I stand sustaining this entire world
with a fragment of my being.

Arjuna, see my forms
in hundreds and thousands;
diverse, divine,
of many colors and shapes.

I am time grown old,
creating world destruction,
set in motion
to annihilate the worlds; …

This form you have seen
is rarely revealed;
the gods are constantly craving
for a vision of this form.

Knowledge is better than practice,
meditation better than knowledge,
rejecting fruits of action
is better still — it brings peace.

The field denotes
this body, and wise men
call one who knows it
the field-knower.

Know me as the field-knower
in all fields — what I deem
to be knowledge is knowledge
of the field and its knower.

Hear from me in summary
what the field is
in its character and changes,
and of the field-knower's power.

Ancient seers have sung of this
in many ways, with varied meters
and with aphorisms on the infinite spirit
laced with logical arguments.

The field contains the great elements,
individuality, understanding,
unmanifest nature, the eleven senses,
and the five sense realms.

Its hands and feet reach everywhere;
its head and face see in every direction;
hearing everything, it remains
in the world, enveloping all.

Lacking all the sense organs,
it shines in their qualities;
unattached, it supports everything;
without qualities, it enjoys them.

Outside and within all creatures,
inanimate but still animate,
too subtle to be known,
it is far distant, yet near.

Undivided, it seems divided
among creatures;
understood as their sustainer,
it devours and creates them.

The light of lights
beyond darkness it is called;
knowledge attained by knowledge,
fixed in the heart of everyone.

So, in summary I have explained
the field and knowledge of it;
a man devoted to me, knowing this,
enters into my being.

Arjuna, know that anything
inanimate or alive with motion
is born from the union
of the field and its knower.

He really sees who sees
that all actions are performed
by nature alone and that the self
is not an actor.

When he perceives the unity
existing in separate creatures
and how they expand from unity,
he attains the infinite spirit.

Just as one sun
illumines this entire world,
so the master of the field
illumines the entire field.

They reach the highest state
who with the eye of knowledge know
the boundary between the knower and its field,
and the freedom creatures have from nature.

Lucidity, passion, dark inertia —
these qualities inherent in nature
bind the unchanging
embodied self in the body.

Lucidity addicts one to joy,
and passion to actions,
but dark inertia obscures knowledge
and addicts one to negligence.

When a man of vision sees
nature's qualities as the agent
of action and knows what lies beyond,
he enters into my being.

Transcending the three qualities
that are the body's source, the self
achieves immortality, freed from the sorrows
of birth, death, and old age.

Lord, what signs mark a man
who passes beyond the three qualities?
What does he do to cross
beyond these qualities?
— Arjuna, Bhagavad Gita, 14.21

He does not dislike light
or activity or delusion;
when they cease to exist
he does not desire them.
— Krishna, Bhagavad Gita, 14.22

He remains disinterested,
unmoved by qualities of nature;
he never wavers, knowing
that only qualities are in motion.

I am the infinite spirit's foundation,
immortal and immutable,
the basis of eternal sacred duty
and of perfect joy.

Roots in the air, branches below,
the tree of life is unchanging,
they say; its leaves are hymns,
and he who knows it knows sacred lore.

Its branches
stretch below and above,
nourished by nature's qualities,
budding with sense objects;
aerial roots
tangled in actions
reach downward
into the world of men.

Its form is unknown
here in the world;
unknown are its end,
its beginning, its extent;
cut down this tree
that has such deep roots
with the sharp ax
of detachment.

Then search to find
the realm
that one enters
without returning:
“I seek refuge
in the original spirit of man,
from which primordial
activity extended.”

A fragment of me in the living world
is the timeless essence of life;
it draws out the senses
and the mind inherent in nature.

I dwell deep
in the heart of everyone;
memory, knowledge,
and reasoning come from me;
I am the object to be known
through all sacred lore;
and I am its knower,
the creator of its final truth.

Demonic men cannot comprehend
activity and rest;
there exists no clarity,
no morality, no truth in them.

They say that the world
has no truth, no basis, no god,
that no power of mutual dependence
is its cause, but only desire.

Mired in this view, lost to themselves
with their meager understanding,
these fiends contrive terrible acts
to destroy the world.

Subject to insatiable desire,
drunk with hypocrisy and pride,
holding false notions from delusion,
they act with impure vows.

In their certainty that life
consists in sating their desires,
they suffer immeasurable anxiety
that ends only with death.

Confused by endless thoughts,
caught in the net of delusion,
given to satisfying their desires,
they fall into hell's foul abyss.

The three gates of hell
that destroy the self
are desire, anger, and greed;
one must relinquish all three.

in judging what to do or avoid;
knowing the norms of tradition,
perform your action here.

Given in due time and place
to a fit recipient
who can give no advantage,
charity is remembered as lucid.

OM TAT SAT: “That Is the Real” —
this is the triple symbol of the infinite spirit
that gave a primordial sanctity
to priests, sacred lore, and sacrifice.

SAT is steadfastness in sacrifice,
in penance, in charity;
any action of this order
is denoted by SAT.

Action in sacrifice, charity,
and penance is to be performed,
not relinquished — for wise men,
they are acts of sanctity.

But even these actions
should be done by relinquishing to me
attachment and the fruit of action —
this is my decisive idea.

Knowledge, its object, and its subject
are the triple stimulus of action;
instrument, act, and agent
are the constituents of action.

Knowledge, action, agent are threefold,
differentiated by qualities of nature;
hear how this has been explained
in the philosophical analysis of qualities.

Know that through lucid knowledge
one sees in all creatures
a single, unchanging existence,
undivided within its divisions.

Action known for its lucidity
is necessary, free of attachment,
performed without attraction or hatred
by one who seeks no fruit.

An agent called pure
has no attachment or individualism,
is resolute and energetic,
unchanged in failure and success.

Listen as I tell you without reserve
about understanding and resolve,
each in three aspects,
according to the qualities of nature.

In one who knows activity and rest,
acts of right and wrong,
bravery and fear, bondage and freedom,
understanding is lucid.

When it sustains acts
of mind, breath, and senses
through discipline without wavering,
resolve is lucid.

Arjuna, now hear about joy,
the three ways of finding delight
through practice
that brings an end to suffering.

The joy of lucidity
at first seems like poison
but is in the end like ambrosia,
from the calm of self-understanding.

The joy that is passion
ateat first seems like ambrosia
when senses encounter sense objects,
but in the end it is like poison.

The joy arising from sleep,
laziness, and negligence,
self-deluding from beginning to end,
is said to be darkly inert.

There is no being on earth
or among the gods in heaven
free from the triad of qualities
that are born of nature.

Better to do one's own duty imperfectly
than to do another man's well;
doing action intrinsic to his being,
a man avoids guilt.

Your resolve is futile
if a sense of individuality
makes you think, “I shall not fight” —
nature will compel you to.

You are bound by your own action,
intrinsic to your being, Arjuna;
even against your will you must do
what delusion now makes you refuse.

Arjuna, the lord resides
in the heart of all creatures,
making them reel magically,
as if a machine moved them.

With your whole being, Arjuna,
take refuge in him alone —
from his grace you will attain
the eternal place that is peace.

This knowledge I have taught
is more arcane than any mystery —
consider it completely,
then act as you choose.

Key Words in the Bhagavad-Gita

[Compiled by the Translator, B.S. Miller, p. 162–168]

Time (kãla) — a word that also means “death”. In Indian thought time is without beginning, endless, all-pervading. In the Gita, Krishna identifies himself as indestructible time that destroys the worlds (10.30, 10.33, 11.32). — B.S. Miller (translator of B-G, p. 168)

## Deletions

It is pleasing to contemplate the whole panorama of differential operations and group actions on a proposition as so many alternate scenes projected through the facets of its differential enlargement onto various perspectives taken up by the viewer.

We hope to work toward a point where the reader finds it pleasing to contemplate the whole panorama of differential operations and group actions on a proposition as just so many facets of its differential enlargement as may be projected onto various perspectives.

Let ${\displaystyle X=\{x_{1},\ldots ,x_{n}\}\!}$ and suppose that ${\displaystyle U\!}$ is any subset of these variables, without loss of generality taking ${\displaystyle U=\{x_{1},\ldots ,x_{j}\},\!}$ for some ${\displaystyle j\in [1,n].\!}$ For convenience, let us give a name to the remaining set of variables, say ${\displaystyle V=\{x_{j+1},\ldots ,x_{n}\}.\!}$

Let ${\displaystyle X=\{x_{1},\ldots ,x_{n}\}\!}$ be our principal set of logical variables, and let ${\displaystyle U\!}$ be any subset of these, assumed without loss of generality to form an initial segment $\displaystyle U = \{ x_1, \ldots, x_j \}\!$ of cardinality $\displaystyle j.\!$ For convenience, let the remaining variables be collected in the set $\displaystyle V = \{ x_{j+1}, \ldots, x_n \}\!$ of cardinality $\displaystyle k = n - j.\!$

Given a proposition whose form of expression $\displaystyle F\!$ is more or less difficult to comprehend it may occur to us that the matter could be simpler if we restrict our attention to a subset of the universe, say one described by the proposition $\displaystyle G,\!$ expressed over the same set of variables as $\displaystyle F.\!$

 $\displaystyle F | G ~=~ \varphi_G F\!$
 $\displaystyle \begin{matrix} \mathbb{B}^j & \to & \mathbb{B}^n & \to & \mathbb{B}^k \\[2pt] (\mathbb{B}^j \to \mathbb{B}) & \to & (\mathbb{B}^n \to \mathbb{B}) & \to & (\mathbb{B}^k \to \mathbb{B}) \end{matrix}$
 $\displaystyle \begin{matrix} U & \to & X & \to & V \\[2pt] (U \to \mathbb{B}) & \to & (X \to \mathbb{B}) & \to & (V \to \mathbb{B}) \end{matrix}$
 $\displaystyle \begin{matrix} \mathbb{B}^j & \to & \mathbb{B}^n & \to & \mathbb{B}^j \times \mathbb{B}^k & \to & && \\[2pt] (\mathbb{B}^j \to \mathbb{B}) & \to & (\mathbb{B}^n \to \mathbb{B}) & \to & (\mathbb{B}^j \times \mathbb{B}^k \to \mathbb{B}) & \to & (\mathbb{B}^j \to (\mathbb{B}^k \to \mathbb{B})) \end{matrix}$

Diffraction = differential factorization (or differential fractionation):

 $\displaystyle \begin{matrix} (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}) & \cong & (\mathbb{D}^n \to (\mathbb{B}^n \to \mathbb{B})) \end{matrix}$

We aim to combine these twin perspectives into a stereoscopic view of the situations represented in propositions.

Between the contrasting perspectives afforded by these two $\displaystyle \iota\!$ 's we have what we need to synthesize a stereologic conception of the situation represented in a proposition.

Between the complementary perspectives afforded by these two $\displaystyle \iota\!$ 's we aim to acquire / struggle to compose a stereoscopic vision of the situations represented in propositions.

Between the complementary perspectives afforded by these two $\displaystyle \iota\!$ 's we hope to fashion / strive to fasten a stereotactic grasp on the situations represented in propositions.

## Document History

Author: Jon Awbrey, August 31, 1995
Course: Engineering 690, Graduate Project
Continued from Winter Term 1995
Supervisors: M.A. Zohdy and F. Mili, Oakland University