In the paper we discuss the following type congruences: $$ \biggl ({np^k\atop mp^k}\biggr ) \equiv \left (m \atop n\right ) \pmod {p^r}, $$ where $p$ is a prime, $n$, $m$, $k$ and $r$ are various positive integers with $n\ge m\ge 1$, $k\ge 1$ and $r\ge 1$. Given positive integers $k$ and $r$, denote by $W(k,r)$ the set of all primes $p$ such that the above congruence holds for every pair of integers $n\ge m\ge 1$. Using Ljunggren's and Jacobsthal's type congruences, we establish several characterizations of sets $W(k,r)$ and inclusion relations between them for various values $k$ and $r$. In particular, we prove that $W(k+i,r)=W(k-1,r)$ for all $k\ge 2$, $i\ge 0$ and $3\le r\le 3k$, and $W(k,r)=W(1,r)$ for all $3\le r\le 6$ and $k\ge 2$. We also noticed that some of these properties may be used for computational purposes related to congruences given above.
The effect of iron deficiency on photosynthetic pigments, ribulose-1,5-bisphosphate carboxylase (RuBPC), and photosystem activities were investigated in field grown grapevine (Vitis vinifera L. cv. Pinot noir) leaves. The contents of chlorophyll (Chl) (a+b) and carotenoids per unit fresh mass showed a progressive decrease upon increase in iron deficiency. Similar results were also observed in content of total soluble proteins and RuBPC activity. The marked loss of large (55 kDa) and small (15 kDa) subunits of RuBPC was also observed in severely chlorotic leaves. However, when various photosynthetic electron transport activities were analysed in isolated thylakoids, a major decrease in the rate of whole chain (H2O → methyl viologen) electron transport was observed in iron deficient leaves. Such reduction was mainly due to the loss of photosystem 2 (PS2) activity. The same results were obtained when Fv/Fm was evaluated by Chl fluorescence measurements in leaves. Smaller inhibition of photosystem 1 (PS1) activity was also observed in both mild and severely chlorotic leaves. The artificial electron donors, diphenyl carbazide and NH2OH, markedly restored the loss of PS2 activity in severely chlorotic leaves. The marked loss of PS2 activity was evidently due to the loss of 33, 23, 28-25, and 17 kDa polypeptides in iron deficient leaves. and M. Bertamini, N. Nedunchezhian, B. Borghi.
Steinhaus [9] prove that if a set $A$ has a positive Lebesgue measure in the line then its distance set contains an interval. He obtained even stronger forms of this result in [9], which are concerned with mutual distances between points in an infinite sequence of sets. Similar theorems in the case we replace distance by mutual ratio were established by Bose-Majumdar [1]. In the present paper, we endeavour to obtain some results related to sets with Baire property in locally compact topological spaces, particular cases of which yield the Baire category analogues of the above results of Steinhaus [9] and their corresponding form for ratios by Bose-Majumdar [1].