Let $$p>5$$ be a fixed prime and assume that $$\alpha _1,\alpha _2,\alpha _3$$ are coprime to p. We study the asymptotic behavior of small solutions congruences form _1x_1^2+\alpha _2x_2^2+\alpha _3x_3^2\equiv 0\bmod {q}$$ with $$q=p^n$$ , where $$\max \{|x_1|,|x_2|,|x_3|\}\le N$$ $$(x_1x_2x_3,p)=1$$ . (In fact, we consider smoothed version this problem.) If $$n\rightarrow \infty $$ establish a...

The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden’s theorem, they proved that if a word avoids Abelian k-powers for some integer k, then its Abelian complexity is unbounded. This suggests the following question: How frequently do Abelian k-powers occur in a word ...

We consider the problem of computing the power of some usual F transforms of the Wilks U, Hotelling-Lawley T, and Pillai V statistics for testing H0: CBA = Œ0 under the multivariate general linear model, Y = XB + ‰ , where the rows of ‰A are taken as independent N(0, Í) random vectors. Keeping all these matrices at full rank, let C be rC × rX and A be P × rA. For determining p-values, Fi (i ∈ {...

Magnetic field (MF) has had many therapeutic usages in traditional medicine. Today, it is used to accelerate the recovery period after surgical operations by some medical centers and in some horse-training farms in the world. There are some scientific reports on the alternative effect of MF on nervous system function, convulsion, pain, neurons, embryonic cells and immune system cells. For this ...