The mathematical genesis of the analytical closed-form solution is presented, and the structure of the theoretical derivation, proofs and numerical results confirm the validity and base principle of the STFT.
Of great significance in Nilakanthan's work includes the inductive mathematical proofs, a derivation and proof of the arctangent trigonometric function, improvements and proofs of other infinite series expansions by Madhava, and in Sanskrit poetry the series π 4 = 1 − 1 3 + 1 5 − 1 7 + 1 9 − 1 11 + ⋯, (5).
The reader can find the detailed derivation and proof in [ 12], from which the validity for t= ℓ networks follows.
Martin-Löf distinguishes carefully between proofs and derivations: a proof object is a witness to the fact that some proposition has been proved; whereas a derivation is the record of the construction of a proof object.
They thus lacked the means for presenting general derivations and proofs of their solution procedures.
Finally derivations, or proofs, of the system, being sequences of formulas, are arithmetized, and are also assigned specific numbers.
In the next section, we give some preliminaries, starting with the discretization of our equation (1.1) is presented, which will be used in later derivations and proofs.
We conclude with a summary and discussion of our results (Section 5), while some technical derivations and proofs are deferred to the Appendices.
Derivations and proofs of the principal relationship and its corollaries as well as examples of analysis and design of bistable arches of arbitrary arch-profiles are presented in the paper.
Thus, if we take the derivation as a proof for the existence of at least one true proposition, we will succumb to an obvious petitio principii.
The derivation and corresponding proof are based on two approaches, which we develop and describe in detail.
