याथाप्रमाण इसके अंग्रेजी अर्थ का उदाहरण

The stationing of Soviet military forces on the islands gave tangible proof of the Soviet threat, and provocative maneuvers by Soviet air and naval forces in Japanese-claimed territory served to reinforce Japanese official policy of close identification with a firm United States-backed posture against Soviet power.

Alternatively, supporters of the stadium have pointed to the awarding of UEFA elite status and the hosting of a number of high-profile matches, most notably the 2002 UEFA Champions League Final and the 2007 UEFA Cup Final, as proof of the renovation's success.

When the municipal grand jury asked him for hard evidence, Parkhurst personally hired a private detective and, with his friend John Erving, went to the streets in disguise to collect proof of the corruption.

The proof of the Einstein relation can be found in many references, for example see Kubo.

The existence of a delta neutral portfolio was shown as part of the original proof of the Black–Scholes model, the first comprehensive model to produce correct prices for some classes of options.

Proof of the Jordan decomposition: For an elementary proof of the existence, uniqueness, and minimality of the Jordan measure decomposition see Fischer (2012).

A proof of the embedding theorem without the vanishing theorem (due to Simon Donaldson) appears in the lecture notes here.

However, in 1987 Pierre Deligne and Luc Illusie gave a purely algebraic proof of the vanishing theorem in .

Using estimates on Kloosterman sums he was able to derive estimates for Fourier coefficients of modular forms in cases where Pierre Deligne's proof of the Weil conjectures was not applicable.

Feige, Goldwasser, Lund, Safra, and Szegedy (1991), and Arora and Safra in 1992 to yield a proof of the PCP theorem by Arora, Lund, Motwani, Sudan, and Szegedy in 1998 .

In 2005 Irit Dinur discovered a significantly simpler proof of the PCP theorem, using expander graphs.