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Differential Logic and Dynamic Systems • Appendices

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Author: Jon Awbrey



Contents

Appendices

Appendix 1. Propositional Forms and Differential Expansions

Table A1. Propositional Forms on Two Variables


       
       


Table A2. Propositional Forms on Two Variables


       
       


Table A3. Ef Expanded Over Differential Features


 


Table A4. Df Expanded Over Differential Features


 


Table A5. Ef Expanded Over Ordinary Features


 


Table A6. Df Expanded Over Ordinary Features


 


Appendix 2. Differential Forms

The actions of the difference operator and the tangent operator on the 16 bivariate propositions are shown in Tables A7 and A8.

Table A7 expands the differential forms that result over a logical basis:

This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive cells of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:

    and    

Table A8 expands the differential forms that result over an algebraic basis:

This set consists of the positive propositions in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the positive differential basis.

Table A7. Differential Forms Expanded on a Logical Basis


 


Table A8. Differential Forms Expanded on an Algebraic Basis


 


Table A9. Tangent Proposition as Pointwise Linear Approximation



Table A10. Taylor Series Expansion Df = df + d²f



Table A11. Partial Differentials and Relative Differentials