Let K be an arbitrary field of characteristic p > 0 and D(Pn) the ring of differential operators on a polynomial algebra Pn in n variables. A long anticipated analogue of the inequality of Bernstein is proved for the ring D(Pn). In fact, three different proofs are given of this inequality (two of which are essentially characteristic free): the first one is based on the concept of the filter dim...

electrochemical behavior of the new mono and dinuclear co (iii)  complexes of  the types trans-[pyco((do)(doh)pn )(l)]pf6   and [{py co((do)(doh)pn)}2 (μ-dicyd)] (pf6)2   have been studied in acetonitrile solution.  all the  mono nuclear complexes  show two quasi- reversible reduction couples associate  to co(iii/ii)and  co(ii/i). each of these redox couples split to two  one electron  waves in...

Electrochemical behavior of the new mono and dinuclear Co (III)  complexes of  the types trans-[pyCo((DO)(DOH)pn )(L)]PF6   and [{py Co((DO)(DOH)pn)}2 (μ-dicyd)] (PF6)2   have been studied in acetonitrile solution.  All the  mono nuclear complexes  show two quasi- reversible reduction couples associate  to Co(III/II)an...

Let K be an arbitrary field of characteristic p > 0 and D(Pn) be the ring of differential operators on a polynomial algebra Pn in n variables. A long anticipated analogue of the inequality of Bernstein is proved for the ring D(Pn). In fact, three different proofs are given of this inequality (two of which are essentially characteristic free): the first one is based on the concept of the filter ...

So, we have (1.2) p1p2 · · · pn < (pn 2 )n (n ≥ 9), where also holds true by computation for 5 ≤ n ≤ 8. In other hand, one can get a trivial lower bound for that product using Euclid’s proof of infinity of primes; Letting En = p1p2 · · · pn−1 for every n ≥ 2, it is clear that pn < En. So, if pn < En < pn+1 then En should has a prime factor among p1, p2, · · · , pn which isn’t possible. Thus En ...