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Cauchy Mean Theorem

100%

Grabowski A.

nr 2

157-166

The purpose of this paper was to prove formally, using the Mizar language, Arithmetic Mean/Geometric Mean theorem known maybe better under the name of AM-GM inequality or Cauchy mean theorem. It states that the arithmetic mean of a list of a non-negative real numbers is greater than or equal to the geometric mean of the same list. The formalization was tempting for at least two reasons: one of them, perhaps the strongest, was that the proof of this theorem seemed to be relatively easy to formalize (e.g. the weaker variant of this was proven in [13]). Also Jensen’s inequality is already present in the Mizar Mathematical Library. We were impressed by the beauty and elegance of the simple proof by induction and so we decided to follow this specific way. The proof follows similar lines as that written in Isabelle [18]; the comparison of both could be really interesting as it seems that in both systems the number of lines needed to prove this are really close. This theorem is item #38 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

O nierówności między średnią arytmetyczną a średnią geometryczną

100%

Charzyński Z. K.

tom z. 26 [199]

165-171

Znana nierówność między średnią arytmetyczną a średnią geometryczną układu (a1,...,an) liczb nieujemnych, tj. nierówność: An(a1,...,an) > Gn(a1,...,an) [gdzie An(a1,.. .,an)=1/n (a1+...+an); Gn(a1,..,an)=(a1*...*an) 1/n), jest często n spotykanym nie tylko w podręcznikach i zbiorach zadań z analizy matematycznej ćwiczeniem (przykładem), mogącym posłużyć także do ilustracji wykorzystania metody indukcji matematycznej przy dowodzeniu nierówności (np. [1],[2], [4], a także [7] ). Mimo to, wciąż pojawiają się zarówno w czasopismach popularnych, jak też naukowych nowe dowody tej "klasycznej" nierówności (np. [3], [5]).

Indices and their applicability under high temperature in potato (Solanum tuberosum L.) cultivars

38%

Paul S. , Gogoi N.

2015

tom 43

The increasing temperature is going to be more vulnerable for cool season crops like potato which requires an optimum productivity temperature of 18 to 20 °C. Thus, breeding for heat tolerance has become very important. Therefore, some previously used indices for abiotic stress tolerance have been used in our study for screening of high temperature stress tolerance in potato. Three high yielding (Kufri jyoti, Kufri megha and Kufri pokraj) and two local (Rangpuria and Badami) commonly grown potato cultivars were selected for our experiment. Potato cultivars were sown under normal condition and two high temperature conditions (polyhouse and early season) and indices such as HSI (heat susceptibility index), HTI (heat tolerance index), GM (geometric mean) and HII (heat intensity index) were used to evaluate the performance of the cultivars under all the three temperature conditions. The positive and significant correlation between HTI (heat tolerance index), and GM (geometric mean) as well as with tuber yield under all the conditions revealed that these indices were efficient in selecting the high temperature tolerant potato cultivars. We recorded the equal applicability of these two indices for both high yielding and local group of potato cultivars. Our study revealed that cultivar Kufri megha and Rangpuria showed higher heat tolerance between high yielding and local cultivars respectively.