Sentence examples for derivation and proof from inspiring English sources

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.

They thus lacked the means for presenting general derivations and proofs of their solution procedures.

We conclude with a summary and discussion of our results (Section 5), while some technical derivations and proofs are deferred to the Appendices.

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.

Within a natural-deduction framework, Tranchini (2012b see Other Internet Resources) has proposed to consider derivations of absurdity as "arguments" rather than "proofs", understanding arguments as senses of derivations and proofs as their denotations, so that a non-normalizable derivation of absurdity is meaningful, but lacks a denotation.

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.

The derivation and corresponding proof are based on two approaches, which we develop and describe in detail.

Our proceeding in these sections provides simultaneously the construction, the derivation, and the proof that the construction always works.

Hence, our proceeding in Sections 2 to 5 provides simultaneously the construction, the derivation, and the proof that the construction always works.

The theoretic derivation and experimental proof of WAL were extensively developed since 1980s, and the investigation on various materials in all dimensions has been a central topic in condensed matter physics for decades.